Question

# A quadratic polynomial, whose zeroes are $-3$ and $4$ is

A

${x}^{2}-x+12$

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B

${x}^{2}+x+12$

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C

$\frac{{x}^{2}}{2}-\frac{x}{2}-6$

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D

$2{x}^{2}+2x-24$

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Solution

## The correct option is C $\frac{{x}^{2}}{2}-\frac{x}{2}-6$Explanation for Correct option: Let $\alpha ,\beta$ be the zeroes of quadratic polynomial such that $\alpha =-3,\beta =4$Sum of zeroes,$\alpha +\beta =-3+4=1$Product of Zeroes,$\alpha \beta =-3×4=-12$Therefore, the quadratic polynomial can be expressed as,$x²-\left(sumofzeroes\right)x+\left(productofzeroes\right)$$\begin{array}{rcl}& =& x²-\left(\alpha +\beta \right)x+\left(\alpha \beta \right)\\ & =& x²–\left(1\right)x+\left(-12\right)\\ & =& x²–x-12\end{array}$On dividing by $2$ we get,$\begin{array}{rcl}& =& \frac{x²}{2}–\frac{x}{2}-\frac{12}{2}\\ & =& \frac{{x}^{2}}{2}-\frac{x}{2}-6\end{array}$Thus, the required polynomial is$\frac{{x}^{2}}{2}-\frac{x}{2}-6$Hence the correct option is (C)

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