Question

If $\alpha$ and $\beta$ are the roots of the equation $a{x}^2+bx+c=0$, where $a\ne 0$, then $(a\alpha +b)(a\beta +b)$ is equal to :

• A. $ab$
• B. $bc$
• C. $ca$
• D. $abc$

Answer 1

The correct option is C $ca$

For the given equation

$a{x}^2+bx+c=0$

We know that for a quadratic equation,

Sum of roots $=\frac{\text{coefficient of}{x}^{}}{\text{coefficient of}{x}^2}$

Product of roots $=\frac{\text{constant term}}{\text{coefficient of}{x}^2}$

Therefore,

$\alpha +\beta =\frac{-b}{a}$ and

$\alpha \beta =\frac{c}{a}$

Now, $(a\alpha +b)(a\beta +b)$

$=a\alpha .a\beta +a\alpha .b+b.a\beta +b.b=a^2\alpha \beta +ab\alpha +ab\beta +b^2=a^2\alpha \beta +ab(\alpha +\beta )+b^2$

On substituting values, we get

$(a\alpha +b)(a\beta +b)$

$=a^2\left(\frac{c}{a}\right)+ab\left(\frac{-b}{a}\right)+b^2=ac-b^2+b^2=ac$

Hence, $(a\alpha +b)(a\beta +b)=ac$