Algebra Examples

$\log_{x + 3} (2 x^{2} + 9) = 2$

Step 1

Rewrite $\log_{x + 3} (2 x^{2} + 9) = 2$ 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 $b^{y} = x$ .

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

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

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

Step 2.1.2

Simplify each term.

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Step 2.1.2.1

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

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

Step 2.1.2.2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 2.1.2.2.1

Apply the distributive property.

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

Step 2.1.2.2.2

Apply the distributive property.

$x \cdot x + x \cdot 3 + 3 (x + 3) - 2 x^{2} = 9$

Step 2.1.2.2.3

Apply the distributive property.

$x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 - 2 x^{2} = 9$

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

Multiply $x$ by $x$ .

$x^{2} + x \cdot 3 + 3 x + 3 \cdot 3 - 2 x^{2} = 9$

Step 2.1.2.3.1.2

Move $3$ to the left of $x$ .

$x^{2} + 3 \cdot x + 3 x + 3 \cdot 3 - 2 x^{2} = 9$

Step 2.1.2.3.1.3

Multiply $3$ by $3$ .

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

Step 2.1.2.3.2

Add $3 x$ and $3 x$ .

$x^{2} + 6 x + 9 - 2 x^{2} = 9$

Step 2.1.3

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

$- x^{2} + 6 x + 9 = 9$

Step 2.2

Subtract $9$ from both sides of the equation.

$- x^{2} + 6 x + 9 - 9 = 0$

Step 2.3

Combine the opposite terms in $- x^{2} + 6 x + 9 - 9$ .

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Step 2.3.1

Subtract $9$ from $9$ .

$- x^{2} + 6 x + 0 = 0$

Step 2.3.2

Add $- x^{2} + 6 x$ and $0$ .

$- x^{2} + 6 x = 0$

Step 2.4

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

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Step 2.4.1

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

$- x \cdot x + 6 x = 0$

Step 2.4.2

Factor $- x$ out of $6 x$ .

$- x \cdot x - x \cdot - 6 = 0$

Step 2.4.3

Factor $- x$ out of $- x (x) - x (- 6)$ .

$- x (x - 6) = 0$

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 $0$ .

$x = 0$

$x - 6 = 0$

Step 2.6

Set $x$ equal to $0$ .

$x = 0$

Step 2.7

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 2.7.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 2.7.2

Add $6$ to both sides of the equation.

$x = 6$

Step 2.8

The final solution is all the values that make $- x (x - 6) = 0$ true.

$x = 0, 6$