Question

Select the correct answer. What are the roots of the quadratic equation Y =x^2-10x+125 A x=5+-5√6 B x=-5+-10i C x=-5+-5√6 D x=5+-10i​

Answer 1

Answer:

Option D: x = 5 ± 10i

Step-by-step explanation:

Given the quadratic equation, y = x² - 10x + 125,

where:

a = 1, b = -10, and c = 125:

Use the following quadratic formula to solve for the roots:

$\displaystyle\mathsf{x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}$

$\displaystyle\mathsf{x=\frac{10\pm\sqrt{(-10)^2-4(1)(125)}}{2(1)}}$

$\displaystyle\mathsf{x=\frac{10\pm\sqrt{100-500}}{2}\:=\:\frac{10\pm\sqrt{-400}}{2}}$

Apply the imaginary unit rule, where it states that:  $\displaystyle\mathsf{\sqrt{-a}\:=\:i\sqrt{a}}$ :  

$\displaystyle\mathsf{x=\frac{10\pm\:20i}{2}}$

$\displaystyle\mathsf{x=\frac{10\:+\:20i}{2}\:=\frac{10(1\:+\:2i)}{2}\:=\:5(1+2i)}\:=5+10i}$

$\displaystyle\mathsf{x=\frac{10\:-\:20i}{2}\:=\frac{10(1\:-\:2i)}{2}\:=\:5(1-2i)}\:=5-10i}$

Therefore, the correct answer is Option D: x = 5 ± 10i .