WILL GIVE BRAINLIEST
Hannah and Han are each trying to solve the equation x² – 8x + 26 = 0. They know that
x = -1 are i& - i, but they are not sure how to use this information to solve for x in their
equation.
Part 1- Here is Hannah's work:
x? - 8x + 26 = 0
X? – 8x = -26
Show Hannah how
to finish her work using completing the square and complex numbers.
Part 2- Han decides to solve the equation using the quadratic
formula. Here is the beginning of his
work
-b+V62-4ac
-(-8)+7-8)2–401|(26)
Finish using the quadratic formula. Simplify the final answer as much as possible.
Answers
Part one:
[tex]x^2-8x=-26[/tex]
Rewrite in the form [tex](x+a)^{2} =b[/tex]
[tex]\left(x-4\right)^2=-10[/tex]
[tex]\mathrm{For\:}f^2\left(x\right)=a\mathrm{\:the\:solutions\:are\:}f\left(x\right)=\sqrt{a},\:-\sqrt{a}[/tex]
Solve [tex]x-4=\sqrt{-10} : x=\sqrt{10} i+4[/tex]
Solve [tex]x-4=\sqrt{-10} : x=-\sqrt{10} i+4[/tex]
[tex]x=\sqrt{10}i+4,\:x=-\sqrt{10}i+4[/tex]
Part two:
[tex]x=\frac{-\left(-8\right)\pm \sqrt{\left(-8\right)^2-4\cdot \:1\cdot \:26}}{2\cdot \:1}[/tex]
Simplify [tex]\sqrt{\left(-8\right)^2-4\cdot \:1\cdot \:26}}: 2\sqrt{10} i[/tex]
[tex]=\frac{-\left(-8\right)\pm \:2\sqrt{10}i}{2\cdot \:1}[/tex]
Separate solutions
[tex]x_1=\frac{-\left(-8\right)+2\sqrt{10}i}{2\cdot \:1},\:x_2=\frac{-\left(-8\right)-2\sqrt{10}i}{2\cdot \:1}[/tex]
[tex]\frac{-(-8)+2\sqrt{10}i }{2*1} :4+\sqrt{10}i[/tex]
[tex]\frac{-(-8)+2\sqrt{10}i }{2*1} :4-\sqrt{10}i[/tex]
[tex]x=4+\sqrt{10}i,\:x=4-\sqrt{10}i[/tex]
The solutions are:-
[tex]x=4+\sqrt{10i}\\\\x=4-\sqrt{10i}[/tex]
What is the equation?
The definition of an equation is a mathematical statement that shows that two mathematical expressions are equal.
Here given equation is
[tex]x^2- 8x + 26 = 0\\\\x^2-8x=-26\\\\(x-4)^2=-10\\\\[/tex]
[tex](x-4)=[/tex]±[tex]\sqrt{-10}[/tex]
[tex]x=\sqrt{10}i+4\\\\x=-\sqrt{10}i+4[/tex]
So,
[tex]x=\frac{-b+-\sqrt{b^2-4ac}}{2a}\\\\x=\frac{8+-\sqrt{(-8)^2-4(1)(26)}}{2(1)}\\\\x=\frac{8+-\sqrt{(-8)^2-4(1)(26)}}{2(1)}\\\\x=\frac{8+-\sqrt{64-104}}{2}\\\\x=\frac{8+-2\sqrt{10i}}{2}\\\\x=\frac{2(4+-\sqrt{10i})}{2}\\\\x=4+\sqrt{10i}\\\\x=4-\sqrt{10i}[/tex]
Hence, the solutions are:-
[tex]x=4+\sqrt{10i}\\\\x=4-\sqrt{10i}[/tex]
To know more about the equation
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Source: i found it in another brainly question which was the same as your question and was expert verified so i hope this helped