Question

Solve each equation by factoring or by using the quadratic formula. The quadratic formula is:

If $a{x}^2+bx+c=0$, with$a\ne 0$, then$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.

$2z^2+7z=0$

Short answer

The values of z are$x=-\frac{7}{2}$ and$z=0$.

Step-by-step solution

- Understand the concept used 

If$a{x}^2+bx+c=0$ , with$a\ne 0$, then $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$. Solving using factoring can be done by splitting the middle term of equation and then making groups and equating them to zero.

- Substitute the values 

It is given that$2z^2+7z=0$.Using quadratic formula$z=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, substitute 2 for a, 7 for b and 0 for c.

Put the values in the formula and get.$z=\frac{-7\pm \sqrt{{\left(7\right)}^2-4\times \left(2\right)\times \left(0\right)}}{2\left(2\right)}$

- Simplify for z 

Simplify the expression for z.

$\begin{array}{c}z=\frac{-\left(7\right)\pm \sqrt{{\left(7\right)}^2-4\times \left(2\right)\times \left(0\right)}}{2\left(2\right)}\\ =\frac{-7\pm \sqrt{49-0}}{4}\\ =\frac{-7\pm \sqrt{49}}{7}\\ =\frac{-7\pm 7}{4}\end{array}$

Therefore, either $z=\frac{-7+7}{4}=\frac{0}{4}$or $z=\frac{-7-7}{4}=\frac{-14}{4}$that is., either$z=0$ or$z=-\frac{7}{2}$ .