Question

If α and β are the roots of the equation \(a{x}^{2} + bx + c=0\). find the equation whose roots are - α and - β.

• A. c$x^2$ - bx + a = 0
• B. a$x^2$ - bx + c = 0
• C. c$x^2$ + bx + a = 0
• D. a$x^2$ + bx + c = 0

Answer 1

The correct option is B

a$x^2$ - bx + c = 0

a$x^2$ + bx + c roots are α and β

α + β = $\frac{-b}{a}$

α . β = $\frac{c}{a}$

When roots are -α and -β

-α -β = -(α + β) = $\frac{b}{a}$

(-α) × (-β) = αβ = $\frac{c}{a}$

Equation is $x^2$ - (-α - β) × + (-α) (-β) = 0

$x^2$ - (+ $\frac{c}{a}$)n + $\frac{c}{a}$ = 0

a$x^2$ - bx + c = 0

Alternatively

If roots are equal in magnitude but opposite in sign we can replace x by (-x) in the given equation.

a${(-x)}^2$ + b(-x) + c = 0

a$x^2$ - bx + c = 0