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Question

Answers

A) ${{x}^{2}}-x+2$

B) ${{x}^{2}}-x-2$

C) ${{x}^{2}}+x-2$

D) None of these

Answer
Verified

Hint: The given question is related to quadratic equations. Try to recall the formulae related to the relation between the coefficients and sum and product of the roots of a quadratic equation.

Complete step-by-step answer:

Before proceeding with the solution, we must know about the relation between the coefficients and sum and product of the roots of the quadratic equation given by \[a{{x}^{2}}+bx+c=0\] .

We know, the roots of the equation \[a{{x}^{2}}+bx+c=0\] are given by the quadratic formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .

Let $\alpha $ and $\beta $ be the roots of the equation. So, $\alpha =\dfrac{-b+\sqrt{{{b}^{2}}-4ac}}{2a}$ and $\beta =\dfrac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}$. The sum of the roots is given as $\alpha +\beta =\left( \dfrac{-b+\sqrt{{{b}^{2}}-4ac}}{2a} \right)+\left( \dfrac{-b-\sqrt{{{b}^{2}}-4ac}}{2a} \right)=\dfrac{-2b}{2a}=\dfrac{-b}{a}$ .

So, the sum of the roots is related to the coefficients as $\alpha +\beta =\dfrac{-b}{a}$ .

The product of the roots is given as $\alpha \beta =\left( \dfrac{-b+\sqrt{{{b}^{2}}-4ac}}{2a} \right)\left( \dfrac{-b-\sqrt{{{b}^{2}}-4ac}}{2a} \right)=\dfrac{{{b}^{2}}-\left( {{b}^{2}}-4ac \right)}{4{{a}^{2}}}=\dfrac{c}{a}$ .

So, the product of the roots is related to the coefficients as $\alpha \beta =\dfrac{c}{a}$ .

Now, we have \[a{{x}^{2}}+bx+c=0\]. On dividing the equation by $a$ , we get ${{x}^{2}}+\dfrac{b}{a}x+\dfrac{c}{a}=0.....(i)$.

We have $\alpha +\beta =\dfrac{-b}{a}$ and $\alpha \beta =\dfrac{c}{a}$ . So, we can rewrite equation \[(i)\] with coefficients in the form sum and product of roots as ${{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta =0$.

Now, coming to the question , we are given the zeros of a quadratic polynomial as $1$ and $-2$. So, the sum of zeroes is equal to $-2+1=-1$ and the product of zeroes is equal to $-2\times 1=-2$ .

Hence, the quadratic polynomial having zeros $1$ and $-2$ is given as ${{x}^{2}}+x-2$ .

Hence , option C. is the correct answer.

Note: The quadratic equation with coefficients in the form sum and product of roots is given as ${{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta =0$ and not ${{x}^{2}}+\left( \alpha +\beta \right)x+\alpha \beta =0$. Students often get confused and make a mistake. Such mistakes should be avoided as they can lead to wrong answers.

Complete step-by-step answer:

Before proceeding with the solution, we must know about the relation between the coefficients and sum and product of the roots of the quadratic equation given by \[a{{x}^{2}}+bx+c=0\] .

We know, the roots of the equation \[a{{x}^{2}}+bx+c=0\] are given by the quadratic formula $x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ .

Let $\alpha $ and $\beta $ be the roots of the equation. So, $\alpha =\dfrac{-b+\sqrt{{{b}^{2}}-4ac}}{2a}$ and $\beta =\dfrac{-b-\sqrt{{{b}^{2}}-4ac}}{2a}$. The sum of the roots is given as $\alpha +\beta =\left( \dfrac{-b+\sqrt{{{b}^{2}}-4ac}}{2a} \right)+\left( \dfrac{-b-\sqrt{{{b}^{2}}-4ac}}{2a} \right)=\dfrac{-2b}{2a}=\dfrac{-b}{a}$ .

So, the sum of the roots is related to the coefficients as $\alpha +\beta =\dfrac{-b}{a}$ .

The product of the roots is given as $\alpha \beta =\left( \dfrac{-b+\sqrt{{{b}^{2}}-4ac}}{2a} \right)\left( \dfrac{-b-\sqrt{{{b}^{2}}-4ac}}{2a} \right)=\dfrac{{{b}^{2}}-\left( {{b}^{2}}-4ac \right)}{4{{a}^{2}}}=\dfrac{c}{a}$ .

So, the product of the roots is related to the coefficients as $\alpha \beta =\dfrac{c}{a}$ .

Now, we have \[a{{x}^{2}}+bx+c=0\]. On dividing the equation by $a$ , we get ${{x}^{2}}+\dfrac{b}{a}x+\dfrac{c}{a}=0.....(i)$.

We have $\alpha +\beta =\dfrac{-b}{a}$ and $\alpha \beta =\dfrac{c}{a}$ . So, we can rewrite equation \[(i)\] with coefficients in the form sum and product of roots as ${{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta =0$.

Now, coming to the question , we are given the zeros of a quadratic polynomial as $1$ and $-2$. So, the sum of zeroes is equal to $-2+1=-1$ and the product of zeroes is equal to $-2\times 1=-2$ .

Hence, the quadratic polynomial having zeros $1$ and $-2$ is given as ${{x}^{2}}+x-2$ .

Hence , option C. is the correct answer.

Note: The quadratic equation with coefficients in the form sum and product of roots is given as ${{x}^{2}}-\left( \alpha +\beta \right)x+\alpha \beta =0$ and not ${{x}^{2}}+\left( \alpha +\beta \right)x+\alpha \beta =0$. Students often get confused and make a mistake. Such mistakes should be avoided as they can lead to wrong answers.

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