Algebra Examples

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

Simplify each term.

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Apply the distributive property.

$\sqrt{2} - x \cdot x - x \cdot - 4 = 7$

Multiply $x$ by $x$ by adding the exponents.

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Move $x$ .

$\sqrt{2} - (x \cdot x) - x \cdot - 4 = 7$

Multiply $x$ by $x$ .

$\sqrt{2} - x^{2} - x \cdot - 4 = 7$

Multiply $- 4$ by $- 1$ .

$\sqrt{2} - x^{2} + 4 x = 7$

Subtract $7$ from both sides of the equation.

$\sqrt{2} - x^{2} + 4 x - 7 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 1$ , $b = 4$ , and $c = \sqrt{2} - 7$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (- 1 \cdot (\sqrt{2} - 7))}}{2 \cdot - 1}$

Simplify.

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Simplify the numerator.

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Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 1 \cdot (\sqrt{2} - 7)}}{2 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$x = \frac{- 4 \pm \sqrt{16 + 4 \cdot (\sqrt{2} - 7)}}{2 \cdot - 1}$

Apply the distributive property.

$x = \frac{- 4 \pm \sqrt{16 + 4 \sqrt{2} + 4 \cdot - 7}}{2 \cdot - 1}$

Multiply $4$ by $- 7$ .

$x = \frac{- 4 \pm \sqrt{16 + 4 \sqrt{2} - 28}}{2 \cdot - 1}$

Subtract $28$ from $16$ .

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

Rewrite $- 12 + 4 \sqrt{2}$ as $2^{2} (- 3 + \sqrt{2})$ .

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Factor $4$ out of $- 12$ .

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

Factor $4$ out of $4 \sqrt{2}$ .

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

Factor $4$ out of $4 (- 3) + 4 (\sqrt{2})$ .

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

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{- 3 + \sqrt{2}}}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{- 4 \pm 2 \sqrt{- 3 + \sqrt{2}}}{- 2}$

Simplify $\frac{- 4 \pm 2 \sqrt{- 3 + \sqrt{2}}}{- 2}$ .

$x = 2 \pm \sqrt{- 3 + \sqrt{2}}$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 1 \cdot (\sqrt{2} - 7)}}{2 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$x = \frac{- 4 \pm \sqrt{16 + 4 \cdot (\sqrt{2} - 7)}}{2 \cdot - 1}$

Apply the distributive property.

$x = \frac{- 4 \pm \sqrt{16 + 4 \sqrt{2} + 4 \cdot - 7}}{2 \cdot - 1}$

Multiply $4$ by $- 7$ .

$x = \frac{- 4 \pm \sqrt{16 + 4 \sqrt{2} - 28}}{2 \cdot - 1}$

Subtract $28$ from $16$ .

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

Rewrite $- 12 + 4 \sqrt{2}$ as $2^{2} (- 3 + \sqrt{2})$ .

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Factor $4$ out of $- 12$ .

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

Factor $4$ out of $4 \sqrt{2}$ .

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

Factor $4$ out of $4 (- 3) + 4 (\sqrt{2})$ .

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

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{- 3 + \sqrt{2}}}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{- 4 \pm 2 \sqrt{- 3 + \sqrt{2}}}{- 2}$

Simplify $\frac{- 4 \pm 2 \sqrt{- 3 + \sqrt{2}}}{- 2}$ .

$x = 2 \pm \sqrt{- 3 + \sqrt{2}}$

Change the $\pm$ to $+$ .

$x = 2 + \sqrt{- 3 + \sqrt{2}}$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Simplify the numerator.

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Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 1 \cdot (\sqrt{2} - 7)}}{2 \cdot - 1}$

Multiply $- 4$ by $- 1$ .

$x = \frac{- 4 \pm \sqrt{16 + 4 \cdot (\sqrt{2} - 7)}}{2 \cdot - 1}$

Apply the distributive property.

$x = \frac{- 4 \pm \sqrt{16 + 4 \sqrt{2} + 4 \cdot - 7}}{2 \cdot - 1}$

Multiply $4$ by $- 7$ .

$x = \frac{- 4 \pm \sqrt{16 + 4 \sqrt{2} - 28}}{2 \cdot - 1}$

Subtract $28$ from $16$ .

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

Rewrite $- 12 + 4 \sqrt{2}$ as $2^{2} (- 3 + \sqrt{2})$ .

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Factor $4$ out of $- 12$ .

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

Factor $4$ out of $4 \sqrt{2}$ .

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

Factor $4$ out of $4 (- 3) + 4 (\sqrt{2})$ .

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

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{- 3 + \sqrt{2}}}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$x = \frac{- 4 \pm 2 \sqrt{- 3 + \sqrt{2}}}{- 2}$

Simplify $\frac{- 4 \pm 2 \sqrt{- 3 + \sqrt{2}}}{- 2}$ .

$x = 2 \pm \sqrt{- 3 + \sqrt{2}}$

Change the $\pm$ to $-$ .

$x = 2 - \sqrt{- 3 + \sqrt{2}}$

The final answer is the combination of both solutions.

$x = 2 + \sqrt{- 3 + \sqrt{2}}, 2 - \sqrt{- 3 + \sqrt{2}}$

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