Basic Math Examples

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

Step 1

Subtract $z$ from both sides of the equation.

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

Step 2

Multiply each term in $\frac{z^{2}}{6} + \frac{z^{3}}{6} - z = 0$ by $6$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{z^{2}}{6} + \frac{z^{3}}{6} - z = 0$ by $6$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Cancel the common factor of $6$ .

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Step 2.2.1.1.1

Cancel the common factor.

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

Step 2.2.1.1.2

Rewrite the expression.

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

Step 2.2.1.2

Cancel the common factor of $6$ .

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Step 2.2.1.2.1

Cancel the common factor.

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

Step 2.2.1.2.2

Rewrite the expression.

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

Step 2.2.1.3

Multiply $6$ by $- 1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $0$ by $6$ .

$z^{2} + z^{3} - 6 z = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $z$ out of $z^{2} + z^{3} - 6 z$ .

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Step 3.1.1

Factor $z$ out of $z^{2}$ .

$z \cdot z + z^{3} - 6 z = 0$

Step 3.1.2

Factor $z$ out of $z^{3}$ .

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

Step 3.1.3

Factor $z$ out of $- 6 z$ .

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

Step 3.1.4

Factor $z$ out of $z \cdot z + z \cdot z^{2}$ .

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

Step 3.1.5

Factor $z$ out of $z (z + z^{2}) + z \cdot - 6$ .

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

Step 3.2

Let $u = z$ . Substitute $u$ for all occurrences of $z$ .

$z (u + u^{2} - 6) = 0$

Step 3.3

Factor $u + u^{2} - 6$ using the AC method.

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Step 3.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 3.3.2

Write the factored form using these integers.

$z ((u - 2) (u + 3)) = 0$

Step 3.4

Factor.

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Step 3.4.1

Replace all occurrences of $u$ with $z$ .

$z ((z - 2) (z + 3)) = 0$

Step 3.4.2

Remove unnecessary parentheses.

$z (z - 2) (z + 3) = 0$

Step 4

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

$z = 0$

$z - 2 = 0$

$z + 3 = 0$

Step 5

Set $z$ equal to $0$ .

$z = 0$

Step 6

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

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Step 6.1

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

$z - 2 = 0$

Step 6.2

Add $2$ to both sides of the equation.

$z = 2$

Step 7

Set $z + 3$ equal to $0$ and solve for $z$ .

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Step 7.1

Set $z + 3$ equal to $0$ .

$z + 3 = 0$

Step 7.2

Subtract $3$ from both sides of the equation.

$z = - 3$

Step 8

The final solution is all the values that make $z (z - 2) (z + 3) = 0$ true.

$z = 0, 2, - 3$