Algebra Examples

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

Step 1

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

$2 x^{2} + 3 x = 6 x + 2$

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 $6 x$ from both sides of the equation.

$2 x^{2} + 3 x - 6 x = 2$

Step 2.1.2

Subtract $6 x$ from $3 x$ .

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

Step 2.2

Subtract $2$ from both sides of the equation.

$2 x^{2} - 3 x - 2 = 0$

Step 2.3

Factor by grouping.

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Step 2.3.1

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 $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = - 3$ .

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Step 2.3.1.1

Factor $- 3$ out of $- 3 x$ .

$2 x^{2} - 3 x - 2 = 0$

Step 2.3.1.2

Rewrite $- 3$ as $1$ plus $- 4$

$2 x^{2} + (1 - 4) x - 2 = 0$

Step 2.3.1.3

Apply the distributive property.

$2 x^{2} + 1 x - 4 x - 2 = 0$

Step 2.3.1.4

Multiply $x$ by $1$ .

$2 x^{2} + x - 4 x - 2 = 0$

Step 2.3.2

Factor out the greatest common factor from each group.

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Step 2.3.2.1

Group the first two terms and the last two terms.

$(2 x^{2} + x) - 4 x - 2 = 0$

Step 2.3.2.2

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

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

Step 2.3.3

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

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

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 $0$ .

$2 x + 1 = 0$

$x - 2 = 0$

Step 2.5

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

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Step 2.5.1

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

$2 x + 1 = 0$

Step 2.5.2

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

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Step 2.5.2.1

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Step 2.5.2.2

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

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Step 2.5.2.2.1

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

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

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $2$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

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

Step 2.5.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 2.5.2.2.3

Simplify the right side.

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Step 2.5.2.2.3.1

Move the negative in front of the fraction.

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

Step 2.6

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

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Step 2.6.1

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

$x - 2 = 0$

Step 2.6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.7

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

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

Step 3

Exclude the solutions that do not make $\log (2 x^{2} + 3 x) = \log (6 x + 2)$ true.

$x = 2$