Find the quadratic equation whose roots are the additive inverses of the roots of the equation $x^{2} - 5 x + 6 = 0$

x^{2} - 6 x + 5 = 0

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x^{2} - 5 x - 6 = 0

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x^{2} + 5 x + 6 = 0

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x^{2} - 5 x + 6 = 0

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Solution

The correct option is C $x^{2} + 5 x + 6 = 0$

First we have to find the roots of the equation $x^{2} - 5 x + 6 = 0$
$x^{2} - 5 x + 6$
$\Rightarrow x^{2} - 2 x - 3 x + 6 = x (x - 2) - 3 (x - 2) \Rightarrow (x - 2) (x - 3) = 0 \Rightarrow (x - 2) = 0 o r (x - 3) = 0 \Rightarrow x = 2, 3$
Now the roots of the required equation are the additive inverses of 2,3. i.e. -2,-3. So the required equation is
$\Rightarrow (x - (- 2)) (x - (- 3)) = 0 \Rightarrow (x + 2) (x + 3) = 0 \Rightarrow x^{2} + 5 x + 6 = 0$
Therefore the required equation is $x^{2} + 5 x + 6 = 0$

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