Question

Choose the quadratic equations whose sum and product of roots are given as $-5$ and $10,$ respectively.

• A. $x^2+5x+10=0$
• B. $x^2+10x+5=0$
• C. $x^2+5x+10=0$
• D. $3x^2+15x+30=0$

Answer 1

Answer: (A), (D)

$3x^2+15x+30=0$

Let $\alpha$ and $\beta$ be the two roots of the required quadratic equation $a{x}^2+bx+c=0,a\ne 0$

Given: $\alpha +\beta =-5,\alpha \beta =10$

We know $\alpha +\beta =-\frac{b}{a}$
$\Rightarrow -\frac{b}{a}=-5$
$\Rightarrow b=5a$

Also, $\alpha \beta =\frac{c}{a}$
$\Rightarrow \frac{c}{a}=10$
$\Rightarrow c=10a$

Substituting these values in the equation we get,
$a{x}^2+5ax+10a=0$
$\Rightarrow a(x^2+5x+10)=0$
This is a family of equations for different values of $a.$

Let's substitute $a=1.$
$\Rightarrow x^2+5x+10=0$

Substituting $a=3.$
$\Rightarrow 3x^2+15x+30=0$

Hence, options (a) and (b) are correct.