Question

If p(x)=ax2+bx+c,then find c/a is equal to

Answer 1

your answer is c/a = product of its zeroes

Answer 2

$\dfrac{c}{a}$ is equal to the product of its zeroes.

Given:

The polynomial $p(x)=ax^{2}+bx+c$

To find:

$\dfrac{c}{a}$ is equal to

Step-by-step explanation:

Let, $\alpha$ and $\beta$ be the zeroes of the given polynomial.

Then, $a\alpha^{2}+b\alpha+c=0$ ... ... (1)

and $a\beta^{2}+b\beta+c=0$ ... ... (2)

Taking (1) - (2), we get

$a(\alpha^{2}-\beta^{2})+b(\alpha-\beta)=0$

$\Rightarrow a(\alpha+\beta)+b=0$, since $(\alpha-\beta)\neq 0$

$\Rightarrow \alpha+\beta=-\dfrac{b}{a}$ ... ... (3)

Now (1) + (2) gives

$a(\alpha^{2}+\beta^{2})+b(\alpha+\beta)+2c=0$

$\Rightarrow a [(\alpha+\beta)^{2}-2\alpha\beta]+b(-\dfrac{b}{a})+2c=0$, by (3)

$\Rightarrow a[(-\dfrac{b}{a})^{2}-2\alpha\beta]-\dfrac{b^{2}}{a}+2c=0$

$\Rightarrow \dfrac{b^{2}}{a}-2a\alpha\beta-\dfrac{b^{2}}{a}+2c=0$

$\Rightarrow 2a\alpha\beta=2c$

$\Rightarrow \dfrac{c}{a}=\alpha\beta$

Thus $\dfrac{c}{a}$ is equal to the product of its zeroes.

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