Basic Math Examples

$\frac{z^{2}}{3} - z = \frac{5}{3}$

Step 1

Subtract $\frac{5}{3}$ from both sides of the equation.

$\frac{z^{2}}{3} - z - \frac{5}{3} = 0$

Step 2

Multiply through by the least common denominator $3$ , then simplify.

Tap for more steps...

Step 2.1

Apply the distributive property.

$3 (\frac{z^{2}}{3}) + 3 (- z) + 3 (- \frac{5}{3}) = 0$

Step 2.2

Simplify.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$3 (\frac{z^{2}}{3}) + 3 (- z) + 3 (- \frac{5}{3}) = 0$

Step 2.2.1.2

Rewrite the expression.

$z^{2} + 3 (- z) + 3 (- \frac{5}{3}) = 0$

Step 2.2.2

Multiply $- 1$ by $3$ .

$z^{2} - 3 z + 3 (- \frac{5}{3}) = 0$

Step 2.2.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 2.2.3.1

Move the leading negative in $- \frac{5}{3}$ into the numerator.

$z^{2} - 3 z + 3 (\frac{- 5}{3}) = 0$

Step 2.2.3.2

Cancel the common factor.

$z^{2} - 3 z + 3 (\frac{- 5}{3}) = 0$

Step 2.2.3.3

Rewrite the expression.

$z^{2} - 3 z - 5 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 1$ , $b = - 3$ , and $c = - 5$ into the quadratic formula and solve for $z$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Step 5

Simplify.

Tap for more steps...

Step 5.1

Simplify the numerator.

Tap for more steps...

Step 5.1.1

Raise $- 3$ to the power of $2$ .

$z = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

Tap for more steps...

Step 5.1.2.1

Multiply $- 4$ by $1$ .

$z = \frac{3 \pm \sqrt{9 - 4 \cdot - 5}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $- 5$ .

$z = \frac{3 \pm \sqrt{9 + 20}}{2 \cdot 1}$

Step 5.1.3

Add $9$ and $20$ .

$z = \frac{3 \pm \sqrt{29}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$z = \frac{3 \pm \sqrt{29}}{2}$

Step 6

The final answer is the combination of both solutions.

$z = \frac{3 + \sqrt{29}}{2}, \frac{3 - \sqrt{29}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$z = \frac{3 + \sqrt{29}}{2}, \frac{3 - \sqrt{29}}{2}$

Decimal Form:

$z = 4.19258240 \dots, - 1.19258240 \dots$