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

Option D: x = 5 ± 10i

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=(-b\pm√(b^2-4ac))/(2a)}`

`\displaystyle\mathsf{x=(10\pm√((-10)^2-4(1)(125)))/(2(1))}`

`\displaystyle\mathsf{x=(10\pm√(100-500))/(2)\:=\:(10\pm√(-400))/(2)}`

Apply the imaginary unit rule, where it states that:
`\displaystyle\mathsf{√(-a)\:=\:i√(a)}` :

`\displaystyle\mathsf{x=(10\pm\:20i)/(2)}`

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

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

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