Algebra Examples

$\sqrt{2} \cdot x^{2} + 7 x + 5 \sqrt{2} = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = \sqrt{2}$ , $b = 7$ , and $c = 5 \sqrt{2}$ into the quadratic formula and solve for $x$ .

$\frac{- 7 \pm \sqrt{7^{2} - 4 \cdot (\sqrt{2} \cdot (5 \sqrt{2}))}}{2 \sqrt{2}}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Raise $7$ to the power of $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 4 \cdot \sqrt{2} \cdot (5 \sqrt{2})}}{2 \sqrt{2}}$

Step 3.1.2

Multiply $5$ by $- 4$ .

$x = \frac{- 7 \pm \sqrt{49 - 20 \sqrt{2} \sqrt{2}}}{2 \sqrt{2}}$

Step 3.1.3

Multiply $- 20 \sqrt{2} \sqrt{2}$ .

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Step 3.1.3.1

Raise $\sqrt{2}$ to the power of $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 20 (\sqrt{2} \sqrt{2})}}{2 \sqrt{2}}$

Step 3.1.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 20 (\sqrt{2} \sqrt{2})}}{2 \sqrt{2}}$

Step 3.1.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{- 7 \pm \sqrt{49 - 20 {\sqrt{2}}^{1 + 1}}}{2 \sqrt{2}}$

Step 3.1.3.4

Add $1$ and $1$ .

$x = \frac{- 7 \pm \sqrt{49 - 20 {\sqrt{2}}^{2}}}{2 \sqrt{2}}$

Step 3.1.4

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

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Step 3.1.4.1

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

$x = \frac{- 7 \pm \sqrt{49 - 20 {(2^{\frac{1}{2}})}^{2}}}{2 \sqrt{2}}$

Step 3.1.4.2

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

$x = \frac{- 7 \pm \sqrt{49 - 20 \cdot 2^{\frac{1}{2} \cdot 2}}}{2 \sqrt{2}}$

Step 3.1.4.3

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

$x = \frac{- 7 \pm \sqrt{49 - 20 \cdot 2^{\frac{2}{2}}}}{2 \sqrt{2}}$

Step 3.1.4.4

Cancel the common factor of $2$ .

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Step 3.1.4.4.1

Cancel the common factor.

$x = \frac{- 7 \pm \sqrt{49 - 20 \cdot 2^{\frac{2}{2}}}}{2 \sqrt{2}}$

Step 3.1.4.4.2

Rewrite the expression.

$x = \frac{- 7 \pm \sqrt{49 - 20 \cdot 2}}{2 \sqrt{2}}$

Step 3.1.4.5

Evaluate the exponent.

$x = \frac{- 7 \pm \sqrt{49 - 20 \cdot 2}}{2 \sqrt{2}}$

Step 3.1.5

Multiply $- 20$ by $2$ .

$x = \frac{- 7 \pm \sqrt{49 - 40}}{2 \sqrt{2}}$

Step 3.1.6

Subtract $40$ from $49$ .

$x = \frac{- 7 \pm \sqrt{9}}{2 \sqrt{2}}$

Step 3.1.7

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 7 \pm \sqrt{3^{2}}}{2 \sqrt{2}}$

Step 3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 7 \pm 3}{2 \sqrt{2}}$

Step 3.2

Multiply $\frac{- 7 \pm 3}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \frac{- 7 \pm 3}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

Multiply $\frac{- 7 \pm 3}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$x = \frac{(- 7 \pm 3) \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 3.3.2

Move $\sqrt{2}$ .

$x = \frac{(- 7 \pm 3) \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 3.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$x = \frac{(- 7 \pm 3) \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 3.3.4

Raise $\sqrt{2}$ to the power of $1$ .

$x = \frac{(- 7 \pm 3) \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 3.3.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{(- 7 \pm 3) \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 3.3.6

Add $1$ and $1$ .

$x = \frac{(- 7 \pm 3) \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Step 3.3.7

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

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Step 3.3.7.1

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

$x = \frac{(- 7 \pm 3) \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Step 3.3.7.2

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

$x = \frac{(- 7 \pm 3) \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 3.3.7.3

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

$x = \frac{(- 7 \pm 3) \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 3.3.7.4

Cancel the common factor of $2$ .

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Step 3.3.7.4.1

Cancel the common factor.

$x = \frac{(- 7 \pm 3) \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 3.3.7.4.2

Rewrite the expression.

$x = \frac{(- 7 \pm 3) \sqrt{2}}{2 \cdot 2}$

Step 3.3.7.5

Evaluate the exponent.

$x = \frac{(- 7 \pm 3) \sqrt{2}}{2 \cdot 2}$

Step 3.4

Multiply $2$ by $2$ .

$x = \frac{(- 7 \pm 3) \sqrt{2}}{4}$

Step 3.5

Factor $- 1$ out of $- 7 \pm 3$ .

$x = \frac{- 1 (- (- 7 \pm 3)) \sqrt{2}}{4}$

Step 3.6

Multiply $- 1$ by $- 1$ .

$x = \frac{1 (- 7 \pm 3) \sqrt{2}}{4}$

Step 3.7

Multiply $- 7 \pm 3$ by $1$ .

$x = \frac{(- 7 \pm 3) \sqrt{2}}{4}$

Step 3.8

Reorder factors in $\frac{(- 7 \pm 3) \sqrt{2}}{4}$ .

$x = \frac{\sqrt{2} (- 7 \pm 3)}{4}$

Step 4

The final answer is the combination of both solutions.

$x = - \sqrt{2}, - \frac{5 \sqrt{2}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \sqrt{2}, - \frac{5 \sqrt{2}}{2}$

Decimal Form:

$x = - 1.41421356 \dots, - 3.53553390 \dots$