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+(b)/(a)x+(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`