Question

If $\alpha +\beta =-2$ and ${\alpha }^3+{\beta }^3=-56$, then the quadratic equation, whose roots are $\alpha$ and $\beta$, is

• A. $x^2+2x–16=0$
• B. $x^2+2x–15=0$
• C. $x^2+2x–12=0$
• D. $x^2+2x–8=0$

Answer 1

The correct option is D

$x^2+2x–8=0$

Explanation for the correct option:

Find the required quadratic equation:

Given, $\alpha +\beta =-2$ and ${\alpha }^3+{\beta }^3=-56$

As we know,

${(\alpha +\beta )}^3={\alpha }^3+{\beta }^3+3\alpha \beta (\alpha +\beta )$

$\Rightarrow$${(-2)}^3=–56+3\left(\alpha \beta \right)(-2)$

$\Rightarrow$ $–8=–56–6\alpha \beta$

$\Rightarrow$ $\alpha \beta =–8$

$\therefore$The required equation will be in the form $x^2–(\alpha +\beta )x+\alpha \beta =0$

$\Rightarrow$ $x^2–(-2)x+(-8)=0$

$\Rightarrow$ ${\mathit{x}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{2}\mathit{x}\mathbf{}\mathbf{–}\mathbf{}\mathbf{8}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}$

Hence, Option ‘D’ is Correct.