Algebra Examples

$\log (x) = \log (2 x^{2} - 2)$

Step 1

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

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

Step 2

Solve for $x$ .

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Step 2.1

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

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

Step 2.2

Add $2$ to both sides of the equation.

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

Step 2.3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.4

Substitute the values $a = - 2$ , $b = 1$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (- 2 \cdot 2)}}{2 \cdot - 2}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

One to any power is one.

$x = \frac{- 1 \pm \sqrt{1 - 4 \cdot - 2 \cdot 2}}{2 \cdot - 2}$

Step 2.5.1.2

Multiply $- 4 \cdot - 2 \cdot 2$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 1 \pm \sqrt{1 + 8 \cdot 2}}{2 \cdot - 2}$

Step 2.5.1.2.2

Multiply $8$ by $2$ .

$x = \frac{- 1 \pm \sqrt{1 + 16}}{2 \cdot - 2}$

Step 2.5.1.3

Add $1$ and $16$ .

$x = \frac{- 1 \pm \sqrt{17}}{2 \cdot - 2}$

Step 2.5.2

Multiply $2$ by $- 2$ .

$x = \frac{- 1 \pm \sqrt{17}}{- 4}$

Step 2.5.3

Simplify $\frac{- 1 \pm \sqrt{17}}{- 4}$ .

$x = \frac{1 \pm \sqrt{17}}{4}$

Step 2.6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{17}}{4}, \frac{1 - \sqrt{17}}{4}$

Step 3

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

$x = \frac{1 + \sqrt{17}}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + \sqrt{17}}{4}$

Decimal Form:

$x = 1.28077640 \dots$