The explicit formula for the arithmetic sequence 12, 5, -2, -9,..is a(n) = -7n + 19.
To find the explicit formula for this arithmetic sequence, we first need to determine the common difference, which is the amount by which each term differs from the previous term.
d = 5 - 12 = -7
d = -2 - 5 = -7
d = -9 - (-2) = -7
Since the common difference is -7, we can use the formula for an arithmetic sequence:
a(n) = a (1) + (n - 1) d
where a(n) is the nth term of the sequence, a (1) is the first term, n is the term number, and d is the common difference.
Plugging in the given values, we get:
a(n) = 12 + (n - 1) (-7)
Simplifying, we get:
a(n) = -7n + 19
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The explicit formula for the arithmetic sequence will be f(n) = 19 -7n.
f(n) = a + (n - 1) d,
where an is the first term and is the explicit formula for an arithmetic series.
d = Common Difference
n = number of terms
The provided arithmetic series is = 12, 5, -2, -9.
First term: a = 12.
Common variation: d= 5-12= 7
[Difference between any two words that follow one another.]
When a = 12 and d = -7 are entered into equation (1), we obtain
f(n) = 12 + (n - 1) ( - 7 )
Thus, the explicit formula for the above arithmetic sequence is
f(n) = 12 = (n - 1). ( - 7 )
If we continue to solve it, we obtain
f(n) = 12 - 7n + 7
f(n) = 19 - 7n
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Which statement is correct? A -5 < 0 because -5 is to the left of 0 on a number line B -5 < 0 because -5 is to the right of 0 on a number line C -5 > 0 because -5 is to the left of 0 on a number line D -5 > 0 because -5 is to the right of 0 on a number line
The correct statement is A -5 < 0 because -5 is to the left of 0 on a number line. Here's a step-by-step explanation:
On a number line, the smaller numbers are to the left and the larger numbers are to the right. Since -5 is to the left of 0, -5 is a smaller number and 0 is a larger number. Therefore, -5 is less than (or <) 0.
So the statement A -5 < 0 because -5 is to the left of 0 on a number line is correct.
In baseball, each time a player attempts to hit the ball, it is recorded. The ratio of hits compared to total attempts is their batting average. Each player on the team wants to have the highest batting average to help their team the most. For the season so far, Jana has hit the ball 8 times out of 10 attempts. Tasha has hit the ball 9 times out of 12 attempts. Which player has a ratio that means they have a better batting average?
Tasha, because she has the lowest ratio since 0.75 < 0.8
Tasha, because she has the highest ratio since 48 over 60 is greater than 45 over 60
Jana, because she has the lowest ratio since 0.75 < 0.8
Jana, because she has the highest ratio since 48 over 60 is greater than 45 over 60
Jana, because she has the highest ratio since 8/10 is greater than 9/12.
What is ratio?A ratio is a comparison of two numbers or quantities expressed in relation to each other. It represents the relative size or magnitude of one quantity with respect to another. Ratios are typically written as a fraction, with the first number being the numerator and the second number being the denominator, and can also be expressed as a decimal or percentage.
What is batting average?Batting average is a statistical measure used in baseball to evaluate a player's performance at the plate. It is calculated as the ratio of a player's total number of hits to their total number of at-bats (the number of times they attempt to hit the ball).
In the given question,
A higher batting average indicates a better performance, since it means the player is successfully hitting the ball more often.
In this case, we are given the number of hits and attempts for two players, Jana and Tasha. To compare their batting averages, we need to calculate the ratio of their hits to their attempts.
Jana has hit the ball 8 times out of 10 attempts, so her batting average is 8/10 = 0.8.
Tasha has hit the ball 9 times out of 12 attempts, so her batting average is 9/12 = 0.75.
To determine which player has the better batting average, we compare their ratios. Since 0.8 is greater than 0.75, Jana has the higher ratio and therefore the better batting average.
So, the answer is Jana, because she has the highest ratio (8/10 = 0.8), which means she has the better batting average compared to Tasha (9/12 = 0.75).
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What is the image of (-2, 6) after a dilation by a scale factor of 1/2 centered at the
origin?
The image of (-2, 6) after a dilation by a scale factor of 1/2 centered at the origin is given as follows:
(-1, 3).
What is a dilation?A dilation is a transformation that changes the size of a figure, but not its shape. Specifically, a dilation is a type of similarity transformation that involves multiplying the coordinates of each point in a figure by a scale factor. This causes the figure to either enlarge or reduce in size.
The scale factor in the context of this problem is given as follows:
1/2.
The coordinates of the original point are given as follows:
(-2, 6).
Multiplying the coordinates of the original point by the scale factor, the coordinates of the image are given as follows:
(-1,3).
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Find the measures of one exterior angle of a regular polygon if the sum of the interior angles is 3960 degrees.
The exteriοr angle οf the pοlygοn is 15°.
What is pοlygοn?A pοlygοn has as many pairs οf interiοr and exteriοr as number οf its sides. Further each pair οf exteriοr and interiοr add up tο 180°.
Sum οf all the exteriοr angles οf any pοlygοn is 360°.
As sum οf all the interiοr angles οf the pοlygοn is 3960°.
Sum οf all pairs οf exteriοr and interiοr angles is
=> 3960°+360°=4320°
As each pair is 180° , Number οf sides οf pοlygοn = 4320/180 = 24
Sum οf exteriοr angles is always 360° and if pοlygοn is regular (i.e. all sides and angles are equal), size οf each exteriοr angle is
=> 360°/24 = 15°
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Simplify the equation
i= square root -1
The equation [tex](\sqrt(6)-i\sqrt(2))/(i\sqrt(3))[/tex] simplifies to 1 - i where numerator and denominator is multiplied by conjugate.
What is conjugate?Conjugate can refer to a pair of complex numbers that differ only in the sign of their imaginary parts. For example, the conjugate of the complex number a + bi is a - bi.
According to question:To simplify this expression, we need to rationalize the denominator, which means getting rid of the square root in the denominator. To do this, we can multiply the numerator and denominator by the conjugate of the denominator, which is -i√3.
[tex](\sqrt(6)-i\sqrt(2))/(i\sqrt(3)) * (-i\sqrt(3))/(-i\sqrt(3))\\= (\sqrt(6)(-i\sqrt(3)) - i\sqrt(2)(-i\sqrt(3))) / (i\sqrt(3)*(-i\sqrt(3)))\\= (-i\sqrt(18) + 3\sqrt(2)) / 3\\= (-i\sqrt(9*2) + 3\sqrt(2)) / 3\\= (-i\sqrt(9) * \sqrt(2) + 3\sqrt(2)) / 3\\= (-3i * \sqrt(2) + 3\sqrt(2)) / 3\\= (3 - 3i) / 3\\= 1 - i[/tex]
Therefore, [tex](\sqrt(6)-i\sqrt(2))/(i\sqrt(3))[/tex] simplifies to 1 - i.
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what is the average time gap between the first cyclists time and each of the remaining cyclists' times (second through fifth) in the 1995 volta a catalunya cycle race if we know the result?
The average time gap between the first cyclist's time and each of the remaining cyclists' times (second through fifth) in the 1995 Volta a Catalunya cycle race is approximately 6 minutes and 7 seconds.
To calculate this, we need to subtract the time of the first cyclist from each of the remaining cyclists' times (second through fifth).The time for the first cyclist was 41:38:33.
The times for the remaining cyclists were as follows:
We can calculate the difference for each cyclist by subtracting the first cyclist's time from their own time:
Adding up all of the times and dividing by four, we get an average of 00:06:07.
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A student thinks of a number. They square it and then subtract 9. Their answer is 315. What number is the student thinking of?
The number that the student is thinking of is 18.
What is number?
A number is an arithmetic value that is used to calculate and represent a quantity. Numerical symbols, such as "3," are written to represent numbers.
Let x be the number that the student is thinking of.
According to the problem, the student squares the number and subtracts 9:
x² - 9 = 315
To solve for x, we can add 9 to both sides of the equation:
x² = 324
Then, we can take the square root of both sides:
x = ±18
Since x must be a positive number (according to the problem statement), we can discard the negative solution. Therefore, the number that the student is thinking of is 18.
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(Find the Smallest Value) Write an application that finds the smallest of several integers. Assume that the first value read specifies the number of values to input from the user.
To find the smallest of several integers, you will need to write a program that takes in a number of values specified by the user as input, and then outputs the smallest value. First, you will need to create a loop that reads in the number of values specified by the user. To do this, you can use a 'for' loop that reads in the values until it reaches the user-specified number of values. Inside the loop, you can create a variable to store the smallest value and compare each value to the smallest value to determine if it is smaller. If the value is smaller, then the smallest value should be updated.
Finally, once the loop has gone through all of the values, the smallest value should be outputted. In summary, to find the smallest value of several integers, you will need to use a 'for' loop to read in the specified number of values from the user, compare each value to the smallest value, and update the smallest value if it is smaller. Then, you can output the smallest value once the loop is complete.
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Use a ruler and pair of compasses to make an accurate drawing of line
AB and its perpendicular bisector, as shown. You must show all of your
construction lines.
Mark point C on your drawing.
Measure the length of AC in your drawing to 1 d.p.
Step-by-step explanation:
1. Draw a line of 8cm.
2. Take a compass and keep it in the length of more than 8 cm.
3. Draw an arc from point A and B which will intersect at point between A and B.
4. Draw a straight line from the arc.
5. You will find out that the line will be drawn exactly between A and B at 4cm.
Identify the slope and y intercept for the equation y = 2x + 3
A rectangle measures 3 2/3 inches by 1/2 inches. What is its area?
the rectangle has a surface area of 11/6 square inches and perimeter is 23/3 inches.
How is area and perimeter determined?We must multiply the length by the width of the rectangle to determine its area. Applying the provided metrics, we have:
Area is equal to the product of length and width.
Area is equal to (11/3) x (1/2) inches.
11/6 square inches is the area.
As a result, the rectangle has a surface area of 11/6 square inches.
The lengths of all four sides must be added up in order to determine the rectangle's perimeter. Applying the provided metrics, we have:
Perimeter equals 2 (length plus width).
Perimeter equals 2 (3 2/3 inches plus 1 inch).
perimeter equals two (11/33 inch plus 1/2 inch).
P = 2 (or 23/6 inches)
Diameter is 23 3/8 inches.
As a result, the rectangle's perimeter is 23/3 inches.
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HW7.4. Find the frequency with the largest amplitude Find the frequency w for which the particular solution to the differential equation dạy dy 2- + dt2 + 2y = eiwt dt has the largest amplitude. You can assume a positive frequency w > 0. Probably the easiest way to do this is to find the particular solution in the form Aeiwt and then minimize the modulus of the denominator of A over all frequencies w. W= number (rtol=0.01, atol=1e-08) ?
Answer:
the frequency with the largest amplitude Find the frequency w for which the particular solution to the differential equation dạy dy 2- + dt2 + 2y = eiwt dt ...
Step-by-step explanation: basically bigger is better
Baseball hats are on sale for 12% off the original price of the sale price is $12.50 what was the original price? round the answer to the nearest cent
The original price of the baseball hat before 12% off in the sale was $14.20.
What is a percentage?A number can be expressed as a fraction of 100 using a percentage. It frequently serves to indicate a portion of a total and is represented by the sign %. We may state that 25% of the class is made up of guys, for instance, if there are 100 pupils in the class and 25 of them are male.
The Roman term per centum, which meaning "by the hundred," is where the word "percent" originates.
Let us suppose the original price of the baseball hat = x.
Given that, baseball hats are on sale for 12% off the original price.
That is,
12.50 = x(1 - 12/100)
12.50 = x(100 - 12)/100
12.50 = x(0.88)
x = 14.2
Hence, the original price of the baseball hat was $14.20.
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out of total student 3/5 are girls .On a particular day one third boys and 2/5 girls were absent. If total absentees was 280 find total number of students
Answer:
Step-by-step explanation:
Calculate fraction of boys
If girls =3/5, then boys = 1 - 3/5 = 2/5.
Calculate fraction of students absent
Girls - 2/5 of 3/5 = 6/25
Boys - 1/3 of 2/5 = 2/15
6/25+2/15=28/75
Calculate total number of students
If 28/75 = 280
280/28=10
10x75=750
Total number of students = 750
Solve for x in the triangle.
Answer:
D. 87
Step-by-step explanation:
You want to know the measure of the angle opposite the longest side in a triangle with side lengths 13, 16, and 20 inches.
Angle relationsThe angle x is opposite the side of length 20 inches in this triangle, which is the longest side. That tells you x is the largest angle.
The largest angle in any triangle is never less than 60°. This eliminates all answer choices except the last one:
x = 87
Law of CosinesIf you want to go to the trouble to solve the triangle, the law of cosines is helpful. For sides a, b, c and angle C, it tells you ...
c² = a² +b² -2ab·cos(C)
Solving for the angle, we have ...
C = arccos((a² +b² -c²)/(2ab))
C = arccos((13² +16² -20²)/(2·13·16)) = arccos(25/416) ≈ 86.55°
x ≈ 87
Drag the term to the corresponding probability. Each term may be used only once. CertainLikelyUnlikelyImpossible Probability Term Probability < 12 Probability = 0 Probability = 1 Probability > 12
Thus, the probability are Zero represents the impossibility.
1/2 equals probable and unlikely
1 = event happens.
Because we don't know how something will turn out, we might talk about the probability of one outcome or the potential for several outcomes.
Typically, the likelihood is stated as the proportion of favourable events to all other potential outcomes in the sample space.
The probability of an event is calculated using the formula P(E) = (Number of favourable outcomes) (Sample space).
Given that probability can only be a number between 0 and 1,
The zeroth value is the impossible.
so there is a 50% chance for both likely and unlikely.
If 1, the event must take place.
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please i need help rn
Therefore, if the relationship between time and distance is proportional and the turtle swims 25 feet in 40 seconds, the table above shows the corresponding distances for other times.
What is proportion?A proportion is a statement that two ratios or fractions are equal. It is a way to express the relationship between two quantities that are proportional to each other. In other words, if we have two ratios that are equal, we can write them as a proportion. Proportions are used in a wide range of fields, including mathematics, science, finance, and engineering. They are often used to make predictions, estimate values, or analyze data.
Here,
To determine the missing values in the table, we can use the fact that the relationship between time and distance is proportional. We can set up a proportion using the initial values given:
25 feet / 40 seconds = d feet / t seconds
Simplifying the left-hand side, we get:
5/8 = d/t
5/8 * 20 = 12.5
Using this proportion, we can fill in the missing values in the table:
Time Distance
(seconds) (feet)
1 5
12.5 10
30 187.5
40 25
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The perimeter of a rectangle is 3.5 times the length. The width is 2.5 cm less than the length. What is the width?
It is 7.5
the steps are in the photo
1. Describe the historical data on Nando’s sales, including a discussion of thegeneral direction of sales and any seasonal tendencies that might beoccurring. 2. Discuss, giving your justifications, which time series forecasting techniquesare appropriate for producing forecasts with this data set. 3. Apply the appropriate forecasting techniques and compare the models basedon ex post forecasts. Choose the best model. 4. Use your chosen forecasting model to generate forecasts for each of themonths in year 2021. 5. Discuss how these forecasts might be integrated into the planning operationsand policy makings in NIH
In Rosettenville, a suburb of Johannesburg, South Africa, Robert Brozin and Fernando Duarte acquired the Chicken Land restaurant in 1987, launching Nando's.
The eatery was renamed Nando's in honor of Fernando. The restaurant incorporated influences from former Mozambican Portuguese colonists, many of whom had relocated to Johannesburg's southeast after their country gained independence in 1975. Expansion was an essential component of their vision from the beginning. Nando's had already grown from one restaurant in 1987 to four by 1990. It became increasingly difficult to implement a common strategy and decision-making became inefficient as new outlets were maintained as separate businesses.
In 1995, Nando's International Holdings (NIH) was established as a new international holding because managing this growingly complex global structure had become extremely challenging. The South African branch of Nando's Group Holdings (NGH) was successfully listed on the Johannesburg Stock Exchange on April 27, 1997. NGH was 54% owned by NIH, with the remaining 26% available to the general public and former joint venture partners. The main goals of the share offer and listing were to broaden the group's capital base and enable group restructuring.
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Find the expected value of the winnings
from a game that has the following
payout probability distribution:
The expected probability distribution is $4.35
We must multiply each supplied chance by the respective payout value in order to determine the expected value.
As a result, we have the payout probability distribution shown below:
0.35(1) + 2(0.2) + 5(0.1) + 8(0.2) + 10(0.15) (0.15)
= 0.35 + 0.4 + 0.5 + 1.6 + 1.5
= $4.35
When the set of possible outcomes is discrete in nature, the distribution is referred to as a discrete probability distribution. When a dice is rolled, all conceivable results, for instance, are discrete and result in a large number of outcomes. A different name for it is the probability mass function.
statistical distributions that can be used
Uniform discrete distribution The chances of each outcome are equal.
Single-trial distribution with two potential results.
A series of Bernoulli events are called the binomial distribution.
Poisson Distribution: The likelihood that a particular occurrence may or may not occur.
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A parking garage has 230 cars in it when it opens at 8???????? (???? = 0). On the interval 0 ≤ ???? ≤ 10, cars enter the parking garage at the rate ???? ′ (????) = 58 cos(0.1635???? − 0.642) cars per hour and cars leave the parking garage at the rate ???? ′ (????) = 65 sin(0.281????) + 7.1 cars per hour (a) How many cars enter the parking garage over the interval ???? = 0 to ???? = 10 hours? (b) Find ????′′(5). Using correct units, explaining the meaning of this value in context of the problem. (c) Find the number of cars in the parking garage at time ???? = 10. Show the work that leads to your answer. (d) Find the time ???? on 0 ≤ ???? ≤ 10, when the number of cars in the parking garage is a maximum. To the nearest whole number, what is the maximum number of cars in the parking garage? Justify your answer
4x 2 +6x−13=3x 2 to the nearest tenth.
The solutions to the equation are x = -4 and x = 1.
What is quadratic formula?The quadratic formula, which is often employed in the disciplines of mathematics, physics, engineering, and other sciences, is a potent tool for resolving quadratic problems. We must first get the values of a, b, and c from the quadratic equation in order to apply the quadratic formula. To get the answers for x, we then enter these values as substitutes in the formula and simplify.
The given equation is 4x² + 6x - 13 = 3x².
Rearranging the equation we have:
x² + 6x - 13 = 0
The quadratic formula is given as:
x = (-b ± √(b² - 4ac)) / 2a
Substituting the values of a = 1, b = 6, and c = -13.
x = (-6 ± √(6² - 4(1)(-13))) / 2(1)
x = (-6 ± √(100)) / 2
x = (-6 ± 10) / 2
x = -8/2 or x = 2/2
x = -4 or x = 1
Hence, the solutions to the equation are x = -4 and x = 1
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is this A.less then
b.equal too
c.more then
Answer:
A. Less than
Step-by-step explanation:
Please help me with my math!!
Answer:
5
Step-by-step explanation:
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To find the current that supplies the maximum wattage, we need to take the derivative of the power function with respect to I, set it equal to zero, and solve for I. The derivative of the power function is:
dW/dI = 120 - 24I
Setting this equal to zero and solving for I, we get:
120 - 24I = 024I = 120I = 5
Therefore, the current that supplies the maximum wattage is 5 amps. We can verify that this is a maximum by checking the second derivative of the power function with respect to I. The second derivative is:
d^2W/dI^2 = -24Since the second derivative is negative, the function has a maximum at I = 5.
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Answer:
We can find the current that supplies the maximum wattage by finding the value of I that maximizes the power function W(I) = 120I - 12I^2.
To do this, we can take the derivative of W with respect to I and set it equal to zero:
dW/dI = 120 - 24I = 0
Solving for I, we get:
I = 5
Therefore, the current that supplies the maximum wattage is 5 amps.
Answer: b. 5
help with number 2 question
Step-by-step explanation:
To make it simple, since the radius is 15, two points on the circle could be (15,0) and (0,-15). Two more straightforward options are (-15,0) or (0,15)
Kate plans to save 5% of her income. She just earned 80.Howmuchmoneyshouldsheputintosavings?
$4
Multiply 80 by 0.05 to get the answer of $4
Answer:
4 (insert the currency needed)
Step-by-step explanation:
To find our answer we have to find 5% of Kate's wages and to do this we have to do 5 divided by 100. Then that answer is multiplied by 80!
5 ÷ 100 = 0.0580 × 0.05 = 4This means she has to put £4 into her savings!
Hope this helps, have a lovely day! :)
a random variable x has the following probability distribution. values of x -1 0 1 probability 0.3 0.4 0.3 (a) calculate the mean of x.
The mean (also called the arithmetic mean or average) is a measure of central tendency that represents the typical or average value of a set of data. The mean is calculated by summing up all the values in the data set and dividing by the number of values.
The mean of x is calculated by the following formula:
mean of x = ∑(x * P(x))
Where, ∑ = Summation operator
x = Value of random variable
P(x) = Probability of the corresponding value of x.
Let's calculate the mean of x using the formula provided above.
mean of x = (-1 × 0.3) + (0 × 0.4) + (1 × 0.3)
= -0.3 + 0 + 0.3
= 0
Therefore, the mean of x is 0.
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when performing regression, why would you want to have a quadratic term? group of answer choices you never want to add a quadratic term when performing regression to better fit a scatterplot with too many outliers to better fit a scatterplot that shows a curve in the data to better fit linear data
So, the right response is that, in order to more accurately fit the scatterplot that depicts a curve in the data, you need add a quadratic component while performing regression.
What is the quadratic term count?As ax² +bx + c, a quadratic equation can be expressed. In a quadratic equation, the largest exponent is 2, which limits the number of terms to a maximum of 3. These terms are exponent 2 (ax²) and exponent 1 (bx) fixed term.
Regression can be improved by including a quadratic component to better match scatterplots of data that display curves. This is so that the independent and dependent variables can have a nonlinear connection, which is made possible by a quadratic term. A quadratic component can assist capture inherent curvature of a data and enhance the fit of a regression model in cases where the connection between both the variables isn't really strictly linear.
A quadratic term may not be appropriate or required, though. A quadratic factor would not increase the model's fit if the variables' relationships are strictly linear and might even result in overfitting. In addition, if the scatterplot contains too many anomalies or the data is not consistent, adding a quadratic factor might not be beneficial.
In order to properly fit a scatter plot graph that depicts a curve in the data, the correct response is you would like to add a quadratic term while performing regression.
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If θ = 1 π 6 , then find exact values for the following: sec ( θ ) equals csc ( θ ) equals tan ( θ ) equals cot ( θ ) equals Add Work
If θ = 1π/6 then six trigonometric functions of θ are: sec(θ), cos(θ), tan(θ), cot(θ), is [tex]((2 \sqrt{(3)})[/tex], [tex]\sqrt(3)/2[/tex], [tex]\sqrt{(3)}/3[/tex], and [tex]\sqrt{(3)[/tex], respectively.
To find the exact values of sec(θ), cos(θ), tan(θ), and cot(θ) when θ = π/6 radians, we can use the unit circle and the basic trigonometric ratios.
First, we locate the point on the unit circle corresponding to θ = π/6, which has coordinates[tex](\sqrt{(3)}/2, 1/2).[/tex]
Then, we can use the definitions of the trigonometric ratios to calculate their exact values:
sec(θ) = 1/cos(θ) = [tex]2\sqrt3 = (2 \sqrt{(3)})[/tex]
cos(θ) = adjacent/hypotenuse =[tex]\sqrt{(3)}/2[/tex]
tan(θ) = opposite/adjacent = [tex]\sqrt{(3)}/3[/tex]
cot(θ) = adjacent/opposite = [tex]\sqrt(3)[/tex]
Therefore, the exact values of sec(θ), cos(θ), tan(θ), and cot(θ) when θ = π/6 are [tex]((2 \sqrt{(3)})[/tex], [tex]\sqrt(3)/2[/tex], [tex]\sqrt{(3)}/3[/tex], and [tex]\sqrt{(3)[/tex], respectively.
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Please help me with my math!
Answer:
-3
Step-by-step explanation:
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To rewrite the quadratic equation 9 = -3x^2 -18x - 25 in the form y = a(x - p)^2 + q, we need to complete the square. First, we factor out the leading coefficient of -3:
-[tex]3(x^2 + 6x + 25/3) = -3(x^2 + 6x + 9 + 16/3)[/tex]
Next, we add and subtract 9 inside the parentheses to complete the square:
[tex]-3(x^2 + 6x + 9 - 9 + 16/3) = -3((x + 3)^2 - 1/3)[/tex]
Simplifying the expression further, we get:
[tex]-3(x + 3)^2 + 3 = y[/tex]
Comparing this to the standard form y = a(x - p)^2 + q, we can see that a = -3, p = -3, and q = 3. Therefore, the value of p is -3.
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Answer:
[tex]p = - 3[/tex]Step-by-step explanation:
To find:-
The value of p .Answer:-
We are here given that a quadratic equation is written in the form of y = a(x-p)² + q and we are interested in finding out the value of "p" .
So , the given quadratic equation to us is ,
[tex]\longrightarrow y = -3x^2-18x-25\\[/tex]
Now complete the square on the RHS side of the equation as ,
Firstly make the coefficient of x² as 1 . This can be done by taking out -3 as common.
[tex]\longrightarrow y = -3\bigg(x^2+6x +\dfrac{25}{3}\bigg)\\[/tex]
we can rewrite it as ,
[tex]\longrightarrow y = -3\bigg( x^2+2(3)(x) +\dfrac{25}{3}\bigg) \\[/tex]
Add and subtract 3² , as ;
[tex]\longrightarrow y = -3\bigg( x^2+2(3)(x) + 3^2-3^2+\dfrac{25}{3}\bigg) \\[/tex]
Rearrange the terms as ,
[tex]\longrightarrow y = -3 \left\{ (x^2+2(3)(x) + 3^2) - 9 +\dfrac{25}{3}\right\} \\[/tex]
Notice the terms inside the small brackets are in the form of [tex]a^2+b^2+2ab[/tex] which is the whole square of [tex] (a+b)[/tex] . Hence , we can write it as ,
[tex]\longrightarrow y =-3\bigg\{ (x+3)^2 + \dfrac{-27+25}{3}\bigg\} \\[/tex]
Simplify,
[tex]\longrightarrow y = -3\bigg\{ (x+3)^2 -\dfrac{2}{3}\bigg\} \\[/tex]
Open the curly brackets by multiplying the terms inside the brackets by -3 as ,
[tex]\longrightarrow y = -3(x+3)^2 - 2 \\[/tex]
Now compare it with [tex] y = a(x-p)^2+q [/tex] . On comparing we get ,
[tex]\longrightarrow \boxed{\boldsymbol{ p =-3}}\\[/tex]
Hence the value of p is -3 .