Basic Math Examples

$a^{2} = \sqrt{2} a + 1$

Step 1

Subtract $\sqrt{2} a$ from both sides of the equation.

$a^{2} - \sqrt{2} a = 1$

Step 2

Subtract $1$ from both sides of the equation.

$a^{2} - \sqrt{2} a - 1 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.1.4

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

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Step 5.1.4.1

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

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

Step 5.1.4.2

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

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

Step 5.1.4.3

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

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

Step 5.1.4.4

Cancel the common factor of $2$ .

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Step 5.1.4.4.1

Cancel the common factor.

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

Step 5.1.4.4.2

Rewrite the expression.

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

Step 5.1.4.5

Evaluate the exponent.

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

Step 5.1.5

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

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Step 5.1.5.1

Multiply $- 4$ by $1$ .

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

Step 5.1.5.2

Multiply $- 4$ by $- 1$ .

$a = \frac{\sqrt{2} \pm \sqrt{2 + 4}}{2 \cdot 1}$

Step 5.1.6

Add $2$ and $4$ .

$a = \frac{\sqrt{2} \pm \sqrt{6}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$a = \frac{\sqrt{2} \pm \sqrt{6}}{2}$

Step 6

The final answer is the combination of both solutions.

$a = \frac{\sqrt{2} + \sqrt{6}}{2}, \frac{\sqrt{2} - \sqrt{6}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{\sqrt{2} + \sqrt{6}}{2}, \frac{\sqrt{2} - \sqrt{6}}{2}$

Decimal Form:

$a = 1.93185165 \dots, - 0.51763809 \dots$