Algebra Examples

$\log_{- 2 x - 3} (3 x^{2} - 18) = 2$

Step 1

Rewrite $\log_{- 2 x - 3} (3 x^{2} - 18) = 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

${(- 2 x - 3)}^{2} = 3 x^{2} - 18$

Step 2

Solve for $x$ .

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Step 2.1

Move all terms containing $x$ to the left side of the equation.

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Step 2.1.1

Subtract $3 x^{2}$ from both sides of the equation.

${(- 2 x - 3)}^{2} - 3 x^{2} = - 18$

Step 2.1.2

Simplify each term.

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Step 2.1.2.1

Rewrite ${(- 2 x - 3)}^{2}$ as $(- 2 x - 3) (- 2 x - 3)$ .

$(- 2 x - 3) (- 2 x - 3) - 3 x^{2} = - 18$

Step 2.1.2.2

Expand $(- 2 x - 3) (- 2 x - 3)$ using the FOIL Method.

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Step 2.1.2.2.1

Apply the distributive property.

$- 2 x (- 2 x - 3) - 3 (- 2 x - 3) - 3 x^{2} = - 18$

Step 2.1.2.2.2

Apply the distributive property.

$- 2 x (- 2 x) - 2 x \cdot - 3 - 3 (- 2 x - 3) - 3 x^{2} = - 18$

Step 2.1.2.2.3

Apply the distributive property.

$- 2 x (- 2 x) - 2 x \cdot - 3 - 3 (- 2 x) - 3 \cdot - 3 - 3 x^{2} = - 18$

Step 2.1.2.3

Simplify and combine like terms.

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Step 2.1.2.3.1

Simplify each term.

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Step 2.1.2.3.1.1

Rewrite using the commutative property of multiplication.

$- 2 \cdot - 2 x \cdot x - 2 x \cdot - 3 - 3 (- 2 x) - 3 \cdot - 3 - 3 x^{2} = - 18$

Step 2.1.2.3.1.2

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

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Step 2.1.2.3.1.2.1

Move $x$ .

$- 2 \cdot - 2 (x \cdot x) - 2 x \cdot - 3 - 3 (- 2 x) - 3 \cdot - 3 - 3 x^{2} = - 18$

Step 2.1.2.3.1.2.2

Multiply $x$ by $x$ .

$- 2 \cdot - 2 x^{2} - 2 x \cdot - 3 - 3 (- 2 x) - 3 \cdot - 3 - 3 x^{2} = - 18$

Step 2.1.2.3.1.3

Multiply $- 2$ by $- 2$ .

$4 x^{2} - 2 x \cdot - 3 - 3 (- 2 x) - 3 \cdot - 3 - 3 x^{2} = - 18$

Step 2.1.2.3.1.4

Multiply $- 3$ by $- 2$ .

$4 x^{2} + 6 x - 3 (- 2 x) - 3 \cdot - 3 - 3 x^{2} = - 18$

Step 2.1.2.3.1.5

Multiply $- 2$ by $- 3$ .

$4 x^{2} + 6 x + 6 x - 3 \cdot - 3 - 3 x^{2} = - 18$

Step 2.1.2.3.1.6

Multiply $- 3$ by $- 3$ .

$4 x^{2} + 6 x + 6 x + 9 - 3 x^{2} = - 18$

Step 2.1.2.3.2

Add $6 x$ and $6 x$ .

$4 x^{2} + 12 x + 9 - 3 x^{2} = - 18$

Step 2.1.3

Subtract $3 x^{2}$ from $4 x^{2}$ .

$x^{2} + 12 x + 9 = - 18$

Step 2.2

Add $18$ to both sides of the equation.

$x^{2} + 12 x + 9 + 18 = 0$

Step 2.3

Add $9$ and $18$ .

$x^{2} + 12 x + 27 = 0$

Step 2.4

Factor $x^{2} + 12 x + 27$ using the AC method.

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Step 2.4.1

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 $27$ and whose sum is $12$ .

$3, 9$

Step 2.4.2

Write the factored form using these integers.

$(x + 3) (x + 9) = 0$

Step 2.5

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

$x + 3 = 0$

$x + 9 = 0$

Step 2.6

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

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Step 2.6.1

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

$x + 3 = 0$

Step 2.6.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.7

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 2.7.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 2.7.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 2.8

The final solution is all the values that make $(x + 3) (x + 9) = 0$ true.

$x = - 3, - 9$