Solve for x log base 4 of 5x^2+2 = log base 4 of x+8

$\log_{4} (5 x^{2} + 2) = \log_{4} (x + 8)$

Step 1

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

$5 x^{2} + 2 = x + 8$

Step 2

Solve for $x$ .

Subtract $x$ from both sides of the equation.

$5 x^{2} + 2 - x = 8$

Subtract $8$ from both sides of the equation.

$5 x^{2} + 2 - x - 8 = 0$

Subtract $8$ from $2$ .

$5 x^{2} - x - 6 = 0$

Factor by grouping.

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 $a \cdot c = 5 \cdot - 6 = - 30$ and whose sum is $b = - 1$ .

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

$5 x^{2} - (x) - 6 = 0$

Rewrite $- 1$ as $5$ plus $- 6$

$5 x^{2} + (5 - 6) x - 6 = 0$

Apply the distributive property.

$5 x^{2} + 5 x - 6 x - 6 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(5 x^{2} + 5 x) - 6 x - 6 = 0$

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

$5 x (x + 1) - 6 (x + 1) = 0$

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

$(x + 1) (5 x - 6) = 0$

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

$x + 1 = 0$

$5 x - 6 = 0$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Set $5 x - 6$ equal to $0$ and solve for $x$ .

Set $5 x - 6$ equal to $0$ .

$5 x - 6 = 0$

Solve $5 x - 6 = 0$ for $x$ .

Add $6$ to both sides of the equation.

$5 x = 6$

Divide each term in $5 x = 6$ by $5$ and simplify.

Divide each term in $5 x = 6$ by $5$ .

$\frac{5 x}{5} = \frac{6}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 x}{5} = \frac{6}{5}$

Divide $x$ by $1$ .

$x = \frac{6}{5}$

The final solution is all the values that make $(x + 1) (5 x - 6) = 0$ true.

$x = - 1, \frac{6}{5}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = - 1, \frac{6}{5}$

Decimal Form:

$x = - 1, 1.2$

Mixed Number Form:

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

Solve for x log base 4 of 5x^2+2 = log base 4 of x+8

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