Algebra Examples

$\frac{1}{3} z^{2} - 4 = - \frac{11}{3}$

Step 1

Combine $\frac{1}{3}$ and $z^{2}$ .

$\frac{z^{2}}{3} - 4 = - \frac{11}{3}$

Step 2

Move all terms not containing $z$ to the right side of the equation.

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Step 2.1

Add $4$ to both sides of the equation.

$\frac{z^{2}}{3} = - \frac{11}{3} + 4$

Step 2.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{z^{2}}{3} = - \frac{11}{3} + 4 \cdot \frac{3}{3}$

Step 2.3

Combine $4$ and $\frac{3}{3}$ .

$\frac{z^{2}}{3} = - \frac{11}{3} + \frac{4 \cdot 3}{3}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{z^{2}}{3} = \frac{- 11 + 4 \cdot 3}{3}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $4$ by $3$ .

$\frac{z^{2}}{3} = \frac{- 11 + 12}{3}$

Step 2.5.2

Add $- 11$ and $12$ .

$\frac{z^{2}}{3} = \frac{1}{3}$

Step 3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$z^{2} = 1$

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$z = \pm \sqrt{1}$

Step 5

Any root of $1$ is $1$ .

$z = \pm 1$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.1

First, use the positive value of the $\pm$ to find the first solution.

$z = 1$

Step 6.2

Next, use the negative value of the $\pm$ to find the second solution.

$z = - 1$

Step 6.3

The complete solution is the result of both the positive and negative portions of the solution.

$z = 1, - 1$