What is quadratic equations by factoring

Answer:

  a.  x²

  b.  144 cm²

Step-by-step explanation:

You want to know an expression for the area of the four white triangles in the given figure, and you want to know that area when x = 12 cm.

If you draw horizontal and vertical lines through the middle of the figure, you can see that for each white triangle there is a matching shaded triangle. That is, the shaded area is equal to the white area.

The shaded area is the square of the length of the side of the shaded square, so is ...

  shaded area = x²

The white area is the same, so is ...

  white area = x²

Using x = 12 cm, the formula from part (a) tells us ...

  white area = (12 cm)² = 144 cm²

The total area of the four white triangles is 144 cm².

__

Additional comment

The relation between the side lengths of the shaded area and the side length of the total area is ...

  2x/√2 = x+6

This has one solution: x = 6/(√2 -1) ≈ 14.485281 (cm).

The figure cannot be drawn to scale for any other value of x, so it is impossible for x=12 cm, unless the outside length is (x+4.97056...)

If you take the white triangles to be isosceles right triangles, each will have an area of A = 1/2bh = 1/2(x/√2)(x/√2) = x²/4. The four of them will have an area of 4(x²/4) = x². If you take the white area to be the overall area less the area of a square of side length x, then the area of the four triangles will be (x+6)² -x² = 12x+36. You will notice that these expressions give different values of the area of the white triangles when x=12. (See the previous comment about the value of x.)

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Answer:

  p:q:r = 12:15:10

Step-by-step explanation:

Given p:q = (2/3):(5/6) and q:r = (3/4):(1/2), you want p:q:r.

We can turn each ratio into a ratio of integers by multiplying by the least common denominator.

  p:q = 4:5 . . . . . . . multiply the given ratio by 6

  q:r = 3:2 . . . . . . . multiply the given ratio by 4

Now, we need to have q be represented in each ratio by the same number. That number will be the least common multiple of its representations in these ratios: LCM(5, 3) = 15.

Multiplying the first ratio by 3, we have ...

  p : q = 12 : 15

Multiplying the second ratio by 5 gives ...

  q : r = 15 : 10

Now, we can write the ratio of interest:

  p : q : r = 12 : 15 : 10

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Answer:

4/3 as an exponent is 4 to the 3rd power ( so four with a three right above it a little)

Step-by-step explanation:

Answer:

Below

Step-by-step explanation:

Point -4, 6    slope = - 3

Point slope form of the line :      ( y-y1) = m (x - x1)

(y-6) = -3 (x - -4)

y - 6 = -3 (x+4)

The midline theorem claims that cutting along the midline of a triangle creates a segment that is parallel to the base and half as long.

The condition for the midline theorem is that the two triangles must have the same size and shape, so all three sides have the same length, and all three angles have the same measure.

For the theorem, we draw a line through two points on either side of the triangle and then we prove that the line divides the sides into equal halves. Once proven that the sides are in two equal halves using congruency it is proved that the base and midline are parallel.

Since the base and midline are parallel to each other using the SAS rule of congruency therefore according to the theorem the segment is half as long as the base.

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So on solving the provided question we can say that here, coefficient of x² in 3x³+ 2x² -x+ 1 is 2

A coefficient in mathematics is a polynomial, a series, or the multiplicative coefficient of a particular term in an expression. Typically numeric, however any expression is permitted. The term "parameter" can also refer to the coefficients themselves if they are variables. A number times a variable equals a coefficient. Coefficient examples include:  The coefficient is 14 in phrase 14c 14c 14c. The coefficient is 1 for word g. multiplies the variable by this amount. example  Given that "z" is a variable and that 6z is the definition of the term, the coefficient is 6. One is the coefficient of the square of x.

here,

coefficient of x² in 3x³+ 2x² -x+ 1 is 2

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Answer:undefined

Step-by-step explanation:

Since ​14​ is equivalent to 8x, that number must be 14 ÷ 8. Then ​14 ÷ 8 = x, because of the inverse relationship between multiplication and division.

There are 4 basic mathematical operations which are addition, subtraction, division and multiplication.

Multiplication and division are inverse operations because multiplication undoes division and also division undoes multiplication.

We want to find the quotient of 14 and 8. This is expressed as 14 ÷ 8.

Now, we are told to consider that some number times 8 equals ​14. Let the number be x and as such we have;

8x = 14

Now, since 14 is equal to 8x, then that number must be gotten by using the inverse relationship by multiplying both sides by 1/8 to get;

x = 14/8

Finally, ​14÷8 = x because of the inverse relationship between multiplication and division.

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Hence option b is the correct option i.e The sample mean's sampling distribution will be roughly normal.

The appropriate response to the question about Sampling Distributions is The sample mean will have a sampling distribution mean that is identical to the population mean.

The sampling distribution must then resemble the normal distribution as the sample size grows.

According to this, if the sample size is large enough, the sampling distribution of the mean will always be normally distributed. The sampling distribution of the mean will be normal regardless of whether the population has a normal, Poisson, binomial, or any other distribution.

A bell-shaped, symmetrical distribution known as a normal distribution has fewer observations the farther out from its center.

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Answer:

8, 3

Step-by-step explanation:

28, 23, 18, 13

Subtract each term from the next term.

23 - 28 = -5

18 - 23 = -5

13 - 18 = -5

All differences are equal. When all differences are equal, you have an arithmetic sequence. The equal differences are called the "common difference." To find a new term, add the common difference to the previous term.

13 + (-5) = 13 - 5 = 8

8 + (-5) = 8 - 5 = 3

Answer: 8, 3

Ann and Byron set up shop 35 meters apart on one side of a creek, with the height of the cliff on the other side of the stream being 106.73 meters.

Since a triangle has three sides and three vertices, it is a polygon. It is a fundamental geometric shape. Triangle ABC is the moniker given to a triangle that has vertices A, B, and C. When the three points are not collinear, a singular plane and triangle are found in Euclidean geometry. A triangle is a polygon if it has three sides and three corners. The points where the three sides meet are known as the triangle's corners.

given

consider the triangles ABC and ACD

from the figure

∠DAC = 36°

∠CAB = 68°

AD = 35 m

we have to find the height of the cliff i.e. he length BC

i.e. the length BC

cosθ = adjacent side / hypotenuse

cos36° = 35 / AC

AC = 43.21 m

tan68° = BC / 43.21

BC = 43.21 * 2.47 = 106.73 m

Ann and Byron set up shop 35 meters apart on one side of a creek, with the height of the cliff on the other side of the stream being 106.73 meters.

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According to the triangle's "angle sum property," a triangle's interior angles add up to 180°.

The smallest polygon with three sides and three interior angles is a triangle.

A triangle is a three-sided polygon that consists of three vertices, three sides, and three angles. Three types of triangles—scalene, isosceles, and equilateral—can be distinguished based on the length of their sides.

It can be an acute-angled, obtuse-angled, or right-angled triangle depending on the size of its angles. A triangle's internal angles add up to 180 degrees. Two more terms—altitude and triangle median—are introduced to you in this essay.

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Yes, 4x is an exponential function. An exponential function is one where there is a constant being raised to a variable.

Exponential functions involves exponents. An exponential function has a constant as its base and a variable as its exponent but not the other way round. if a function has a variable as the base and a constant as the exponent then it is a power function but not an exponential function. The exponential function is a type of mathematical function which are helpful in finding the growth or decay of population, money, price that are growing or decay exponentially. A basic exponential function from its definition is of the form f(x) = b x, where 'b' is a constant and 'x' is a variable. generally it refers to the positive valued function of a real variable. it can be extended to the complex numbers or generalized to other mathematical objects like matrices.

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The distance between Afia's school and her babysitter's house is 16 units.

Distance is a measurement of how far away two things or locations are, either numerically or occasionally qualitatively. Distance can refer to a physical length in physics or to an estimate based on other factors in ordinary language.

The distance between two points in a coordinate plane can be found using the distance formula:

d = √((x2 - x1)^2 + (y2 - y1)^2)

Where (x1, y1) and (x2, y2) are the coordinates of the two points, and d is the distance between them.

In this case, the coordinates of Afia's school are (-8, -3) and the coordinates of her babysitter's house are (8, -3).

Plugging these values into the distance formula:

d = √((8 - (-8))^2 + (-3 - (-3))^2)

d = √((8 + 8)^2 + (0)^2)

d = √(16^2)

d = √(256)

d = 16 units

Therefore, the distance between Afia's school and her babysitter's house is 16 units.

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If in rectangular park that is 63 yards by 16 yards, a dog began in the northwest corner and ran south along the length of the park and Finally, the dog ran back to the northwest corner. The dog ran 65 yards.

Distance can be defined as how far an object is from another object.

Using the Pythagoreans theorem to find the distance

c² = √a² + b²

Let plug in the formula

c² = √63² + 16²

c² = √3,969 + 256

c = √4,225

c = 65 yards

Therefore the distance is 65 yards.

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Answer: y = 1/4x +1

Step-by-step explanation:

First, we can find the midpoint of the line segment using the average of x and y coordinates of the endpoints:

M(x,y) = ((-2 + (-6))/2 , (-9 + 7)/2)

M(x,y) = (-4, -1)

To find the slope of the perpendicular bisector, we have to take the negative reciprocal of the slope of the line segment which is (y2-y1)/(x2-x1) = (7-(-9))/(-6-(-2)) = -16/4 = -4

Slope of the perpendicular bisector = -1/slope of the line segment = -1/(-4) = 1/4

Now we can use the point-slope form to find the equation of the line:

y - y1 = m(x - x1)

y - (-1) = 1/4(x + 4)

y = 1/4x +1

so the equation of the perpendicular bisector of the line segment whose endpoints are (-2,-9) and (-6,7) is y = 1/4x +1

Answer:

Option D

Step-by-step explanation:

Remember that the sum of all angles in ANY triangle is 180 degrees. Also, remember that an acute triangle has angles that are all less than 90 degrees. So not only do our options have to add up to 180 degrees, but they also have to be less than 90 degrees in order for the triangle to be acute.

We can immediately exclude options A, B, and C because all three options have angles greater than 90 degrees, which means the triangle isn't acute.

Your answer is option D or "70, 80, 30." Since...

70 + 80 + 30 = 180
70 < 90
80 < 90
30 < 90

Hope this helps.

After integrating the expression  [tex]$$I=\int_1^{\sqrt{3}} 5 \arctan \left(\frac{1}{x}\right) d x$$[/tex] we will get the value as: [tex]$I=\left(\frac{5 \sqrt{3}}{6}-\frac{5}{4}\right)^\pi+\frac{5}{2} \ln |2|$[/tex]

Integration by parts, also known as partial integration, is a technique used in calculus and more widely in mathematical analysis to determine the integral of a function's product in terms of the integral of the product of the function's derivative and antiderivative.

The integration of the result of two functions is the integration by parts. Typically, the two functions are shown as f(x) and g. (x). The first function, f(x), is chosen from the two functions in such a way that its derivative formula exists, while the second function, g(x), is chosen in such a way that its integral formula exists.

∫ f(x).g(x).dx = f(x) ∫ g(x).dx - ∫(f'(x) ∫g(x).dx).dx + C

We have, [tex]$$I=\int_1^{\sqrt{3}} 5 \arctan \left(\frac{1}{x}\right) d x$$[/tex]

We know that

[tex]\frac{1}{x}=arccot(x) \\I=5 \int_1^{\sqrt{3}} arccot(x) dx[/tex]

Integrating by parts

[tex]$$\int f g^{\prime}=f g-\int f^{\prime} g$$[/tex]

let  us assume that,

[tex]$\quad f=\cot ^{-1}(x) \quad g^{\prime}=1$[/tex]

[tex]$I=5 \int_1^{\sqrt{3}} \cot ^{-1}(x) d x$[/tex]

[tex]$I=5\left[x \cot ^{-1}(x)-\int \frac{-x}{x^2+1} d x\right]$[/tex]

[tex]$I=5\left[x \cot ^{-1}(x)+\frac{1}{2} \ln \left|x^2+1\right|\right]_1^{\sqrt{3}}$[/tex]

[tex]$I=5\left[\left(\sqrt{3} \cot ^{-1}(\sqrt{3})+\frac{1}{2} \ln |4|\right)-\left(\cot ^{-1}(1)+\frac{1}{2} \ln |2|\right)\right]$[/tex]

[tex]$I=5\left[\frac{\sqrt{3} \pi}{6}+\ln 2-\frac{\pi}{4}-\frac{1}{2} \ln |2|\right]$[/tex]

[tex]$I=\left(\frac{5 \sqrt{3}}{6}-\frac{5}{4}\right)^\pi+\frac{5}{2} \ln |2|$[/tex]

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The correct question should be:

Evaluate the integral [tex]$$I=\int_1^{\sqrt{3}} 5 \arctan \left(\frac{1}{x}\right) d x$$[/tex]

Answer:

m∠1 = 32

Step-by-step explanation:

∠WXZ = 115

m∠2 = m∠1 + 51

x + x + 51 = 115

2x + 51 = 115

2x = 64

x = 32

The surface area of the square pyramid is B)61.25[tex]cm^2[/tex]

What is surface area of square pyramid?

The term "square pyramid" also refers to a polyhedron having a square base and four triangular faces. Where the four faces come together is known as the apex. The pyramid's faces connect the base and the tip. The calculation for the square pyramid's surface area adds the areas of the four triangular faces and the base.

Here the given measures are base = 3.5 cm and height = 7cm.

Now using surface area formula then

Surface area of the square pyramid = Base area + 2bl square unit.

Base area = 3.5 × 3.5 = 12.25[tex]cm^2[/tex]

=> Surface area = 12.25+ 2×3.5×7

=> Surface area = 12.25+49

=> Surface area = 61.25[tex]cm^2[/tex]

Hence the surface area of the square pyramid is B)61.25[tex]cm^2[/tex]

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It is possible to construct a triangle with lengths of its sides 8cm, 7cm and 4cm because the sum of two sides of a triangle is greater than the third side.

Now, According to the question:

We know that:

What is triangle inequality rule?

The triangle inequality theorem states that for any given triangle, the sum of the two sides is always greater than the  third side. The Triangle is a polygon with three line segments as its boundaries.

Suppose A, B and C are three sides of a triangle, then by using triangle inequality rule, A + B > C then A, B and C make a triangle.

We have the sides:

4 , 7 and 8

4 + 7 > 8

It is possible to construct a triangle with lengths of its sides 8cm, 7cm and 4cm because the sum of two sides of a triangle is greater than the third side.

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Answer:

Step-by-step explanation: v = 1/3π[tex]r^{2}[/tex]h

[tex]\frac{1}{3}[/tex]π([tex]1.5^{2}[/tex]) x 15

[tex]\frac{1}{3}[/tex]π x 2.25 x 15

=35.34291

The cost of admission for an adult is $10 and for a child is $6

The cost of each ride for an adult is $1.50 and for a child is $1

The adult will spend $10 for admission + $1.5 x 7 rides = $17.5

The child will spend $6 for admission + $1 x 7 rides = $13

The adult spent $17.5 - $13 = $4.5 more than the child.

Answer:

Step-by-step explanation:

Answer:20

Step-by-step explanation: 5+5+5+5

Answer:

4 times 5 is equal to 20.

Step-by-step explanation:

By finding the unit rates for the two jars, we will see that the better option is the 28-ounce jar.

To find wich option is the better buy, we need to find the unit rate for each of these jars.

To get the unit rate, we need to take the quotient between the cost of each jar and the volume of peanut butter that each jar has.

We know that the jar of 28 ounces has a cost of $4.29, then its unit rate is:

U =  ($4.29)/28 oz = $0.153 per oz.

This means that with this jar, each ounce costs $0.153

For the 16 ounce jar, the cost is $2.49, then the unit rate is:

U' = $2.49/16 oz = $0.156 per oz.

So here the cost per ounce is larger, then the better option is the 28-ounce jar.

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he system of equations used to represent this scenario is y = –x + 10 and y = 4x, where x and y are the two numbers.

To find the larger number, we can solve the system of equations by setting y = 4x in the first equation and solving for x.

y = –x + 10

4x = –x + 10

3x = 10

x = 10/3

So the smaller number is 10/3. To find the larger number, we know that it is 4 times the smaller number, which is 4(10/3) = 40/3.

Therefore, the larger number is 40/3.

Answer:

The larger number = 8

Step-by-step explanation:

Below is the system of equations:

y = -x + 10

y = 4x

Since both equations equal "y", we can set them equal to each other:

4x = -x + 10

4x + x = 10

5x = 10

x = 2              

This means that at x = 2, the y-values of both equations are the same and is when the criteria of "the sum of two numbers are 10 and the larger number is 4 times the smaller number" is met

"x" is the smaller number. To find the bigger number, input "x" into either one of the equations in the system:

y = 4(2) = 8

The given statement "If point (x, y) is reflected over the x-axis, the resulting point is (-x, y)" is true.

When you reflect a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is taken to be the additive inverse. The reflection of point (x, y) across the x-axis is (x, -y).

The given statement is "If point (x, y) is reflected over the x-axis, the resulting point is (-x, y)".

When you reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate is taken to be the additive inverse. The reflection of point (x, y) across the y-axis is (-x, y).

Therefore, the given statement is true.

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Answer:

The y-intercept of -5x+7y=8 is (0,8/7)

Step-by-step explanation:

The y-intercept of a linear equation is the point where the graph of the equation crosses the y-axis. In this equation, when x = 0, the equation becomes 7y = 8, so the y-intercept is (0, 8/7).

Answer:I hope this helps

Step-by-step explanation:

-5x+7y=8

To find the -intercept, substitute x=0 something like this

-5 times 0+7y=8

Solve the equation for Y which is something like

y=8/7 this your answer for the y intercept.