Solve for x log base x+6 of 2x^2-4x=2

$\log_{x + 6} (2 x^{2} - 4 x) = 2$

Rewrite $\log_{x + 6} (2 x^{2} - 4 x) = 2$ 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 $b^{y} = x$ .

${(x + 6)}^{2} = 2 x^{2} - 4 x$

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 x^{2} - 4 x = {(x + 6)}^{2}$

Simplify ${(x + 6)}^{2}$ .

Rewrite ${(x + 6)}^{2}$ as $(x + 6) (x + 6)$ .

$2 x^{2} - 4 x = (x + 6) (x + 6)$

Expand $(x + 6) (x + 6)$ using the FOIL Method.

Apply the distributive property.

$2 x^{2} - 4 x = x (x + 6) + 6 (x + 6)$

Apply the distributive property.

$2 x^{2} - 4 x = x \cdot x + x \cdot 6 + 6 (x + 6)$

Apply the distributive property.

$2 x^{2} - 4 x = x \cdot x + x \cdot 6 + 6 x + 6 \cdot 6$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$2 x^{2} - 4 x = x^{2} + x \cdot 6 + 6 x + 6 \cdot 6$

Move $6$ to the left of $x$ .

$2 x^{2} - 4 x = x^{2} + 6 \cdot x + 6 x + 6 \cdot 6$

Multiply $6$ by $6$ .

$2 x^{2} - 4 x = x^{2} + 6 x + 6 x + 36$

Add $6 x$ and $6 x$ .

$2 x^{2} - 4 x = x^{2} + 12 x + 36$

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

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

$2 x^{2} - 4 x - x^{2} = 12 x + 36$

Subtract $12 x$ from both sides of the equation.

$2 x^{2} - 4 x - x^{2} - 12 x = 36$

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

$x^{2} - 4 x - 12 x = 36$

Subtract $12 x$ from $- 4 x$ .

$x^{2} - 16 x = 36$

Subtract $36$ from both sides of the equation.

$x^{2} - 16 x - 36 = 0$

Factor $x^{2} - 16 x - 36$ using the AC method.

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 $- 16$ .

$- 18, 2$

Write the factored form using these integers.

$(x - 18) (x + 2) = 0$

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

$x - 18 = 0$

$x + 2 = 0$

Set $x - 18$ equal to $0$ and solve for $x$ .

Set $x - 18$ equal to $0$ .

$x - 18 = 0$

Add $18$ to both sides of the equation.

$x = 18$

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

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

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $(x - 18) (x + 2) = 0$ true.

$x = 18, - 2$

Solve for x log base x+6 of 2x^2-4x=2

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