Algebra Examples

$\log_{3} (2 x^{2} + 7) - \log_{3} (3 x + 6) = 1$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{3} (\frac{2 x^{2} + 7}{3 x + 6}) = 1$

Step 1.2

Factor $3$ out of $3 x + 6$ .

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Step 1.2.1

Factor $3$ out of $3 x$ .

$\log_{3} (\frac{2 x^{2} + 7}{3 (x) + 6}) = 1$

Step 1.2.2

Factor $3$ out of $6$ .

$\log_{3} (\frac{2 x^{2} + 7}{3 x + 3 \cdot 2}) = 1$

Step 1.2.3

Factor $3$ out of $3 x + 3 \cdot 2$ .

$\log_{3} (\frac{2 x^{2} + 7}{3 (x + 2)}) = 1$

Step 2

Rewrite $\log_{3} (\frac{2 x^{2} + 7}{3 (x + 2)}) = 1$ 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 $b^{y} = x$ .

$3^{1} = \frac{2 x^{2} + 7}{3 (x + 2)}$

Step 3

Cross multiply to remove the fraction.

$2 x^{2} + 7 = 3 (3 (x + 2))$

Step 4

Simplify $3 (3 (x + 2))$ .

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Step 4.1

Apply the distributive property.

$2 x^{2} + 7 = 3 (3 x + 3 \cdot 2)$

Step 4.2

Multiply $3$ by $2$ .

$2 x^{2} + 7 = 3 (3 x + 6)$

Step 4.3

Apply the distributive property.

$2 x^{2} + 7 = 3 (3 x) + 3 \cdot 6$

Step 4.4

Multiply.

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Step 4.4.1

Multiply $3$ by $3$ .

$2 x^{2} + 7 = 9 x + 3 \cdot 6$

Step 4.4.2

Multiply $3$ by $6$ .

$2 x^{2} + 7 = 9 x + 18$

Step 5

Subtract $9 x$ from both sides of the equation.

$2 x^{2} + 7 - 9 x = 18$

Step 6

Factor by grouping.

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Step 6.1

Reorder terms.

$2 x^{2} - 9 x + 7 = 18$

Step 6.2

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 $a \cdot c = 2 \cdot 7 = 14$ and whose sum is $b = - 9$ .

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Step 6.2.1

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

$2 x^{2} - 9 x + 7 = 18$

Step 6.2.2

Rewrite $- 9$ as $- 2$ plus $- 7$

$2 x^{2} + (- 2 - 7) x + 7 = 18$

Step 6.2.3

Apply the distributive property.

$2 x^{2} - 2 x - 7 x + 7 = 18$

Step 6.3

Factor out the greatest common factor from each group.

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Step 6.3.1

Group the first two terms and the last two terms.

$(2 x^{2} - 2 x) - 7 x + 7 = 18$

Step 6.3.2

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

$2 x (x - 1) - 7 (x - 1) = 18$

Step 6.4

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

$(x - 1) (2 x - 7) = 18$

Step 7

Simplify $(x - 1) (2 x - 7)$ .

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Step 7.1

Expand $(x - 1) (2 x - 7)$ using the FOIL Method.

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Step 7.1.1

Apply the distributive property.

$x (2 x - 7) - 1 (2 x - 7) = 18$

Step 7.1.2

Apply the distributive property.

$x (2 x) + x \cdot - 7 - 1 (2 x - 7) = 18$

Step 7.1.3

Apply the distributive property.

$x (2 x) + x \cdot - 7 - 1 (2 x) - 1 \cdot - 7 = 18$

Step 7.2

Simplify and combine like terms.

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Step 7.2.1

Simplify each term.

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Step 7.2.1.1

Rewrite using the commutative property of multiplication.

$2 x \cdot x + x \cdot - 7 - 1 (2 x) - 1 \cdot - 7 = 18$

Step 7.2.1.2

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

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Step 7.2.1.2.1

Move $x$ .

$2 (x \cdot x) + x \cdot - 7 - 1 (2 x) - 1 \cdot - 7 = 18$

Step 7.2.1.2.2

Multiply $x$ by $x$ .

$2 x^{2} + x \cdot - 7 - 1 (2 x) - 1 \cdot - 7 = 18$

Step 7.2.1.3

Move $- 7$ to the left of $x$ .

$2 x^{2} - 7 \cdot x - 1 (2 x) - 1 \cdot - 7 = 18$

Step 7.2.1.4

Multiply $2$ by $- 1$ .

$2 x^{2} - 7 x - 2 x - 1 \cdot - 7 = 18$

Step 7.2.1.5

Multiply $- 1$ by $- 7$ .

$2 x^{2} - 7 x - 2 x + 7 = 18$

Step 7.2.2

Subtract $2 x$ from $- 7 x$ .

$2 x^{2} - 9 x + 7 = 18$

Step 8

Subtract $18$ from both sides of the equation.

$2 x^{2} - 9 x + 7 - 18 = 0$

Step 9

Subtract $18$ from $7$ .

$2 x^{2} - 9 x - 11 = 0$

Step 10

Factor by grouping.

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Step 10.1

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 $a \cdot c = 2 \cdot - 11 = - 22$ and whose sum is $b = - 9$ .

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Step 10.1.1

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

$2 x^{2} - 9 x - 11 = 0$

Step 10.1.2

Rewrite $- 9$ as $2$ plus $- 11$

$2 x^{2} + (2 - 11) x - 11 = 0$

Step 10.1.3

Apply the distributive property.

$2 x^{2} + 2 x - 11 x - 11 = 0$

Step 10.2

Factor out the greatest common factor from each group.

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Step 10.2.1

Group the first two terms and the last two terms.

$(2 x^{2} + 2 x) - 11 x - 11 = 0$

Step 10.2.2

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

$2 x (x + 1) - 11 (x + 1) = 0$

Step 10.3

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

$(x + 1) (2 x - 11) = 0$

Step 11

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

$x + 1 = 0$

$2 x - 11 = 0$

Step 12

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

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Step 12.1

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

$x + 1 = 0$

Step 12.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 13

Set $2 x - 11$ equal to $0$ and solve for $x$ .

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Step 13.1

Set $2 x - 11$ equal to $0$ .

$2 x - 11 = 0$

Step 13.2

Solve $2 x - 11 = 0$ for $x$ .

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Step 13.2.1

Add $11$ to both sides of the equation.

$2 x = 11$

Step 13.2.2

Divide each term in $2 x = 11$ by $2$ and simplify.

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Step 13.2.2.1

Divide each term in $2 x = 11$ by $2$ .

$\frac{2 x}{2} = \frac{11}{2}$

Step 13.2.2.2

Simplify the left side.

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Step 13.2.2.2.1

Cancel the common factor of $2$ .

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Step 13.2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{11}{2}$

Step 13.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{11}{2}$

Step 14

The final solution is all the values that make $(x + 1) (2 x - 11) = 0$ true.

$x = - 1, \frac{11}{2}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$x = - 1, \frac{11}{2}$

Decimal Form:

$x = - 1, 5.5$

Mixed Number Form:

$x = - 1, 5 \frac{1}{2}$