Algebra Examples

$\frac{z^{2}}{3} = \frac{z}{2} + \frac{5}{6}$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $\frac{z}{2}$ from both sides of the equation.

$\frac{z^{2}}{3} - \frac{z}{2} = \frac{5}{6}$

Step 1.2

Subtract $\frac{5}{6}$ from both sides of the equation.

$\frac{z^{2}}{3} - \frac{z}{2} - \frac{5}{6} = 0$

Step 2

To write $\frac{z^{2}}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{z^{2}}{3} \cdot \frac{2}{2} - \frac{z}{2} - \frac{5}{6} = 0$

Step 3

To write $- \frac{z}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{z^{2}}{3} \cdot \frac{2}{2} - \frac{z}{2} \cdot \frac{3}{3} - \frac{5}{6} = 0$

Step 4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{z^{2}}{3}$ by $\frac{2}{2}$ .

$\frac{z^{2} \cdot 2}{3 \cdot 2} - \frac{z}{2} \cdot \frac{3}{3} - \frac{5}{6} = 0$

Step 4.2

Multiply $3$ by $2$ .

$\frac{z^{2} \cdot 2}{6} - \frac{z}{2} \cdot \frac{3}{3} - \frac{5}{6} = 0$

Step 4.3

Multiply $\frac{z}{2}$ by $\frac{3}{3}$ .

$\frac{z^{2} \cdot 2}{6} - \frac{z \cdot 3}{2 \cdot 3} - \frac{5}{6} = 0$

Step 4.4

Multiply $2$ by $3$ .

$\frac{z^{2} \cdot 2}{6} - \frac{z \cdot 3}{6} - \frac{5}{6} = 0$

Step 5

Combine the numerators over the common denominator.

$\frac{z^{2} \cdot 2 - z \cdot 3}{6} - \frac{5}{6} = 0$

Step 6

Combine the numerators over the common denominator.

$\frac{z^{2} \cdot 2 - z \cdot 3 - 5}{6} = 0$

Step 7

Simplify each term.

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Step 7.1

Move $2$ to the left of $z^{2}$ .

$\frac{2 \cdot z^{2} - z \cdot 3 - 5}{6} = 0$

Step 7.2

Multiply $3$ by $- 1$ .

$\frac{2 z^{2} - 3 z - 5}{6} = 0$

Step 8

Factor by grouping.

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Step 8.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 5 = - 10$ and whose sum is $b = - 3$ .

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Step 8.1.1

Factor $- 3$ out of $- 3 z$ .

$\frac{2 z^{2} - 3 z - 5}{6} = 0$

Step 8.1.2

Rewrite $- 3$ as $2$ plus $- 5$

$\frac{2 z^{2} + (2 - 5) z - 5}{6} = 0$

Step 8.1.3

Apply the distributive property.

$\frac{2 z^{2} + 2 z - 5 z - 5}{6} = 0$

Step 8.2

Factor out the greatest common factor from each group.

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Step 8.2.1

Group the first two terms and the last two terms.

$\frac{(2 z^{2} + 2 z) - 5 z - 5}{6} = 0$

Step 8.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{2 z (z + 1) - 5 (z + 1)}{6} = 0$

Step 8.3

Factor the polynomial by factoring out the greatest common factor, $z + 1$ .

$\frac{(z + 1) (2 z - 5)}{6} = 0$

Step 9

Set the numerator equal to zero.

$(z + 1) (2 z - 5) = 0$

Step 10

Solve the equation for $z$ .

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Step 10.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$z + 1 = 0$

$2 z - 5 = 0$

Step 10.2

Set $z + 1$ equal to $0$ and solve for $z$ .

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Step 10.2.1

Set $z + 1$ equal to $0$ .

$z + 1 = 0$

Step 10.2.2

Subtract $1$ from both sides of the equation.

$z = - 1$

Step 10.3

Set $2 z - 5$ equal to $0$ and solve for $z$ .

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Step 10.3.1

Set $2 z - 5$ equal to $0$ .

$2 z - 5 = 0$

Step 10.3.2

Solve $2 z - 5 = 0$ for $z$ .

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Step 10.3.2.1

Add $5$ to both sides of the equation.

$2 z = 5$

Step 10.3.2.2

Divide each term in $2 z = 5$ by $2$ and simplify.

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Step 10.3.2.2.1

Divide each term in $2 z = 5$ by $2$ .

$\frac{2 z}{2} = \frac{5}{2}$

Step 10.3.2.2.2

Simplify the left side.

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Step 10.3.2.2.2.1

Cancel the common factor of $2$ .

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Step 10.3.2.2.2.1.1

Cancel the common factor.

$\frac{2 z}{2} = \frac{5}{2}$

Step 10.3.2.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{5}{2}$

Step 10.4

The final solution is all the values that make $(z + 1) (2 z - 5) = 0$ true.

$z = - 1, \frac{5}{2}$