Algebra Examples

$\log_{4} (9 x) = \log_{4} (x^{2} - 36)$

Step 1

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$9 x = x^{2} - 36$

Step 2

Solve for $x$ .

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Step 2.1

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

$9 x - x^{2} = - 36$

Step 2.2

Add $36$ to both sides of the equation.

$9 x - x^{2} + 36 = 0$

Step 2.3

Factor the left side of the equation.

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Step 2.3.1

Factor $- 1$ out of $9 x - x^{2} + 36$ .

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Step 2.3.1.1

Reorder $9 x$ and $- x^{2}$ .

$- x^{2} + 9 x + 36 = 0$

Step 2.3.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 9 x + 36 = 0$

Step 2.3.1.3

Factor $- 1$ out of $9 x$ .

$- (x^{2}) - (- 9 x) + 36 = 0$

Step 2.3.1.4

Rewrite $36$ as $- 1 (- 36)$ .

$- (x^{2}) - (- 9 x) - 1 \cdot - 36 = 0$

Step 2.3.1.5

Factor $- 1$ out of $- (x^{2}) - (- 9 x)$ .

$- (x^{2} - 9 x) - 1 \cdot - 36 = 0$

Step 2.3.1.6

Factor $- 1$ out of $- (x^{2} - 9 x) - 1 (- 36)$ .

$- (x^{2} - 9 x - 36) = 0$

Step 2.3.2

Factor.

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Step 2.3.2.1

Factor $x^{2} - 9 x - 36$ using the AC method.

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Step 2.3.2.1.1

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 $- 36$ and whose sum is $- 9$ .

$- 12, 3$

Step 2.3.2.1.2

Write the factored form using these integers.

$- ((x - 12) (x + 3)) = 0$

Step 2.3.2.2

Remove unnecessary parentheses.

$- (x - 12) (x + 3) = 0$

Step 2.4

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

$x - 12 = 0$

$x + 3 = 0$

Step 2.5

Set $x - 12$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 12$ equal to $0$ .

$x - 12 = 0$

Step 2.5.2

Add $12$ to both sides of the equation.

$x = 12$

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

The final solution is all the values that make $- (x - 12) (x + 3) = 0$ true.

$x = 12, - 3$

Step 3

Exclude the solutions that do not make $\log_{4} (9 x) = \log_{4} (x^{2} - 36)$ true.

$x = 12$