Question

3. Write a quadratic equation such thatthe sum of its roots is -4 and theproduct of its roots is 24.

Answer 1

In a quadratic equation such as:

$ax^2+bx+c=0$

We can see that its roots are independent from the coefficient a, because:

$\begin{gathered} ax^2+bx+c=0 \\ x^2+\frac{b}{a}x+\frac{c}{a}=0 \\ x^2+Bx+C=0 \end{gathered}$

Another thing we know if that we can rewrite an quadratic equation using its roots by:

$a(x-r_1)(x-r_2)$

Which makes the linear coefficient B the same as the negative of the sum of the roots and the coefficien C the same as the product of the roots:

$\begin{gathered} a(x-r_1)(x-r_2)=a(x^2-(r_1+r_2)x+r_1r_2)=a(x^2+Bx+C) \\ B=-(r_1+r_2) \\ C=r_1r_2 \end{gathered}$

Thus, since we can choose the value of a, lets use a = 1 to make it simpler.

This makes:sum

$\begin{gathered} B=-(r_1+r_2)=-(-4)=4 \\ C=r_1r_2=24 \end{gathered}$

And the quadratic equations becomes:

$x^2+4x+24=0$