Basic Math Examples

$180 = z (z - 5)$

Step 1

Rewrite the equation as $z (z - 5) = 180$ .

$z (z - 5) = 180$

Step 2

Simplify $z (z - 5)$ .

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Step 2.1

Apply the distributive property.

$z \cdot z + z \cdot - 5 = 180$

Step 2.2

Simplify the expression.

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Step 2.2.1

Multiply $z$ by $z$ .

$z^{2} + z \cdot - 5 = 180$

Step 2.2.2

Move $- 5$ to the left of $z$ .

$z^{2} - 5 z = 180$

Step 3

Subtract $180$ from both sides of the equation.

$z^{2} - 5 z - 180 = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = 1$ , $b = - 5$ , and $c = - 180$ into the quadratic formula and solve for $z$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (1 \cdot - 180)}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 5$ to the power of $2$ .

$z = \frac{5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 180}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot - 180$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$z = \frac{5 \pm \sqrt{25 - 4 \cdot - 180}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $- 180$ .

$z = \frac{5 \pm \sqrt{25 + 720}}{2 \cdot 1}$

Step 6.1.3

Add $25$ and $720$ .

$z = \frac{5 \pm \sqrt{745}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$z = \frac{5 \pm \sqrt{745}}{2}$

Step 7

The final answer is the combination of both solutions.

$z = \frac{5 + \sqrt{745}}{2}, \frac{5 - \sqrt{745}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$z = \frac{5 + \sqrt{745}}{2}, \frac{5 - \sqrt{745}}{2}$

Decimal Form:

$z = 16.14734406 \dots, - 11.14734406 \dots$