Algebra Examples

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

Step 1

Simplify the left side.

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Step 1.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{3} (\frac{2 x^{2} + 2}{x + 3}) = 0$

Step 1.2

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

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Step 1.2.1

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

$\log_{3} (\frac{2 (x^{2}) + 2}{x + 3}) = 0$

Step 1.2.2

Factor $2$ out of $2$ .

$\log_{3} (\frac{2 (x^{2}) + 2 (1)}{x + 3}) = 0$

Step 1.2.3

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

$\log_{3} (\frac{2 (x^{2} + 1)}{x + 3}) = 0$

Step 2

Rewrite $\log_{3} (\frac{2 (x^{2} + 1)}{x + 3}) = 0$ 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 $b^{y} = x$ .

$3^{0} = \frac{2 (x^{2} + 1)}{x + 3}$

Step 3

Cross multiply to remove the fraction.

$2 (x^{2} + 1) = 3^{0} (x + 3)$

Step 4

Simplify $3^{0} (x + 3)$ .

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Step 4.1

Anything raised to $0$ is $1$ .

$2 (x^{2} + 1) = 1 (x + 3)$

Step 4.2

Multiply $x + 3$ by $1$ .

$2 (x^{2} + 1) = x + 3$

Step 5

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

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Step 5.1

Subtract $x$ from both sides of the equation.

$2 (x^{2} + 1) - x = 3$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$2 x^{2} + 2 \cdot 1 - x = 3$

Step 5.2.2

Multiply $2$ by $1$ .

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

Step 6

Move all terms not containing $x$ to the right side of the equation.

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Step 6.1

Subtract $2$ from both sides of the equation.

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

Step 6.2

Subtract $2$ from $3$ .

$2 x^{2} - x = 1$

Step 7

Subtract $1$ from both sides of the equation.

$2 x^{2} - x - 1 = 0$

Step 8

Factor by grouping.

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Step 8.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = - 1$ .

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Step 8.1.1

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

$2 x^{2} - (x) - 1 = 0$

Step 8.1.2

Rewrite $- 1$ as $1$ plus $- 2$

$2 x^{2} + (1 - 2) x - 1 = 0$

Step 8.1.3

Apply the distributive property.

$2 x^{2} + 1 x - 2 x - 1 = 0$

Step 8.1.4

Multiply $x$ by $1$ .

$2 x^{2} + x - 2 x - 1 = 0$

Step 8.2

Factor out the greatest common factor from each group.

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Step 8.2.1

Group the first two terms and the last two terms.

$(2 x^{2} + x) - 2 x - 1 = 0$

Step 8.2.2

Factor out the greatest common factor (GCF) from each group.

$x (2 x + 1) - (2 x + 1) = 0$

Step 8.3

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$(2 x + 1) (x - 1) = 0$

Step 9

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

$2 x + 1 = 0$

$x - 1 = 0$

Step 10

Set $2 x + 1$ equal to $0$ and solve for $x$ .

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Step 10.1

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Step 10.2

Solve $2 x + 1 = 0$ for $x$ .

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Step 10.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 10.2.2

Divide each term in $2 x = - 1$ by $2$ and simplify.

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Step 10.2.2.1

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 10.2.2.2

Simplify the left side.

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Step 10.2.2.2.1

Cancel the common factor of $2$ .

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Step 10.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Step 10.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Step 10.2.2.3

Simplify the right side.

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Step 10.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Step 11

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 11.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 11.2

Add $1$ to both sides of the equation.

$x = 1$

Step 12

The final solution is all the values that make $(2 x + 1) (x - 1) = 0$ true.

$x = - \frac{1}{2}, 1$

Step 13

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{1}{2}, 1$

Decimal Form:

$x = - 0.5, 1$