Question

Solve the following polynomial equation using formula method

3x²+9x+7=0

Solve the following polynomial equation using factorization method

X³+2x²‐9x‐18=0

Answer 1

Let us solve the polynomial equation $3x^2+9x+7=0$ using formula method. It follows that $D=9^2-4\cdot 3\cdot 7=81-84=-3.$ Therefore, the equation has no real solutions. The complex solutions are the following:

$x_1=\frac{-9+i\sqrt{3}}{6}$ and $x_2=\frac{-9-i\sqrt{3}}{6}.$

Let us solve the polynomial equation $x^3+2x^2-9x-18=0$ using factorization method. It follows that

$x^3+2x^2-9x-18=x^2(x+2)-9(x+2)=(x^2-9)(x+2)=(x-3)(x+3)(x+2),$

and hence the equation $x^3+2x^2-9x-18=0$ is equivalent to $(x-3)(x+3)(x+2)=0.$

Therefore, it has the following solutions:

$x_1=3,x_2=-3$ and $x_3=-2.$