If α and β are the roots of the equation a x2 + bx + c. find the equation whose roots are - α and - β.
ax2 - bx + c = 0
Solution: ax2 + bx + c roots are α and β
α + β = −ba
α . β = ca
When roots are -α and -β
-α -β = -(α + β) = ba
(-α) × (-β) = αβ = ca
Equation is x2 - (-α - β) × + (-α) (-β) = 0
x2 - (+ ca)n + ca = 0
ax2 - 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
ax2 - bx + c = 0