Algebra Examples

$x^{2} - \sqrt{2} x - 12$

Step 1

Set $x^{2} - \sqrt{2} x - 12$ equal to $0$ .

$x^{2} - \sqrt{2} x - 12 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 1$ , $b = - \sqrt{2}$ , and $c = - 12$ into the quadratic formula and solve for $x$ .

$\frac{\sqrt{2} \pm \sqrt{{(- \sqrt{2})}^{2} - 4 \cdot (1 \cdot - 12)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Apply the product rule to $- \sqrt{2}$ .

$x = \frac{\sqrt{2} \pm \sqrt{{(- 1)}^{2} {\sqrt{2}}^{2} - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Step 4.1.2

Raise $- 1$ to the power of $2$ .

$x = \frac{\sqrt{2} \pm \sqrt{1 {\sqrt{2}}^{2} - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Step 4.1.3

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$x = \frac{\sqrt{2} \pm \sqrt{{\sqrt{2}}^{2} - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Step 4.1.4

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 4.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \frac{\sqrt{2} \pm \sqrt{{(2^{\frac{1}{2}})}^{2} - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Step 4.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \frac{\sqrt{2} \pm \sqrt{2^{\frac{1}{2} \cdot 2} - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Step 4.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$x = \frac{\sqrt{2} \pm \sqrt{2^{\frac{2}{2}} - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Step 4.1.4.4

Cancel the common factor of $2$ .

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Step 4.1.4.4.1

Cancel the common factor.

$x = \frac{\sqrt{2} \pm \sqrt{2^{\frac{2}{2}} - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Step 4.1.4.4.2

Rewrite the expression.

$x = \frac{\sqrt{2} \pm \sqrt{2 - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Step 4.1.4.5

Evaluate the exponent.

$x = \frac{\sqrt{2} \pm \sqrt{2 - 4 \cdot 1 \cdot - 12}}{2 \cdot 1}$

Step 4.1.5

Multiply $- 4 \cdot 1 \cdot - 12$ .

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Step 4.1.5.1

Multiply $- 4$ by $1$ .

$x = \frac{\sqrt{2} \pm \sqrt{2 - 4 \cdot - 12}}{2 \cdot 1}$

Step 4.1.5.2

Multiply $- 4$ by $- 12$ .

$x = \frac{\sqrt{2} \pm \sqrt{2 + 48}}{2 \cdot 1}$

Step 4.1.6

Add $2$ and $48$ .

$x = \frac{\sqrt{2} \pm \sqrt{50}}{2 \cdot 1}$

Step 4.1.7

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 4.1.7.1

Factor $25$ out of $50$ .

$x = \frac{\sqrt{2} \pm \sqrt{25 (2)}}{2 \cdot 1}$

Step 4.1.7.2

Rewrite $25$ as $5^{2}$ .

$x = \frac{\sqrt{2} \pm \sqrt{5^{2} \cdot 2}}{2 \cdot 1}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{\sqrt{2} \pm 5 \sqrt{2}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{\sqrt{2} \pm 5 \sqrt{2}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\sqrt{2} \pm 5 \sqrt{2}}{2}$

Decimal Form:

$x = 4.24264068 \dots, - 2.82842712 \dots$