Pre-Algebra Examples

$6 x (3 x + 9) = 180$

Step 1

Divide each term in $6 x (3 x + 9) = 180$ by $6$ and simplify.

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Step 1.1

Divide each term in $6 x (3 x + 9) = 180$ by $6$ .

$\frac{6 x (3 x + 9)}{6} = \frac{180}{6}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify terms.

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Step 1.2.1.1

Cancel the common factor of $6$ .

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Step 1.2.1.1.1

Cancel the common factor.

$\frac{6 x (3 x + 9)}{6} = \frac{180}{6}$

Step 1.2.1.1.2

Divide $x (3 x + 9)$ by $1$ .

$x (3 x + 9) = \frac{180}{6}$

Step 1.2.1.2

Apply the distributive property.

$x (3 x) + x \cdot 9 = \frac{180}{6}$

Step 1.2.1.3

Reorder.

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Step 1.2.1.3.1

Rewrite using the commutative property of multiplication.

$3 x \cdot x + x \cdot 9 = \frac{180}{6}$

Step 1.2.1.3.2

Move $9$ to the left of $x$ .

$3 x \cdot x + 9 \cdot x = \frac{180}{6}$

Step 1.2.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.2.1

Move $x$ .

$3 (x \cdot x) + 9 \cdot x = \frac{180}{6}$

Step 1.2.2.2

Multiply $x$ by $x$ .

$3 x^{2} + 9 \cdot x = \frac{180}{6}$

$3 x^{2} + 9 x = \frac{180}{6}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $180$ by $6$ .

$3 x^{2} + 9 x = 30$

Step 2

Subtract $30$ from both sides of the equation.

$3 x^{2} + 9 x - 30 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $3$ out of $3 x^{2} + 9 x - 30$ .

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Step 3.1.1

Factor $3$ out of $3 x^{2}$ .

$3 (x^{2}) + 9 x - 30 = 0$

Step 3.1.2

Factor $3$ out of $9 x$ .

$3 (x^{2}) + 3 (3 x) - 30 = 0$

Step 3.1.3

Factor $3$ out of $- 30$ .

$3 x^{2} + 3 (3 x) + 3 \cdot - 10 = 0$

Step 3.1.4

Factor $3$ out of $3 x^{2} + 3 (3 x)$ .

$3 (x^{2} + 3 x) + 3 \cdot - 10 = 0$

Step 3.1.5

Factor $3$ out of $3 (x^{2} + 3 x) + 3 \cdot - 10$ .

$3 (x^{2} + 3 x - 10) = 0$

Step 3.2

Factor.

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Step 3.2.1

Factor $x^{2} + 3 x - 10$ using the AC method.

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Step 3.2.1.1

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 $- 10$ and whose sum is $3$ .

$- 2, 5$

Step 3.2.1.2

Write the factored form using these integers.

$3 ((x - 2) (x + 5)) = 0$

Step 3.2.2

Remove unnecessary parentheses.

$3 (x - 2) (x + 5) = 0$

Step 4

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

$x - 2 = 0$

$x + 5 = 0$

Step 5

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

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Step 5.1

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

$x - 2 = 0$

Step 5.2

Add $2$ to both sides of the equation.

$x = 2$

Step 6

Set $x + 5$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Step 6.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 7

The final solution is all the values that make $3 (x - 2) (x + 5) = 0$ true.

$x = 2, - 5$