Precalculus Examples

${(2 x - 1)}^{2} = 8$

Step 1

Subtract $8$ from both sides of the equation.

${(2 x - 1)}^{2} - 8 = 0$

Step 2

Simplify ${(2 x - 1)}^{2} - 8$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(2 x - 1)}^{2}$ as $(2 x - 1) (2 x - 1)$ .

$(2 x - 1) (2 x - 1) - 8 = 0$

Step 2.1.2

Expand $(2 x - 1) (2 x - 1)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$2 x (2 x - 1) - 1 (2 x - 1) - 8 = 0$

Step 2.1.2.2

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 1 - 1 (2 x - 1) - 8 = 0$

Step 2.1.2.3

Apply the distributive property.

$2 x (2 x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1 - 8 = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1 - 8 = 0$

Step 2.1.3.1.2

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

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Step 2.1.3.1.2.1

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1 - 8 = 0$

Step 2.1.3.1.2.2

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1 - 8 = 0$

Step 2.1.3.1.3

Multiply $2$ by $2$ .

$4 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1 - 8 = 0$

Step 2.1.3.1.4

Multiply $- 1$ by $2$ .

$4 x^{2} - 2 x - 1 (2 x) - 1 \cdot - 1 - 8 = 0$

Step 2.1.3.1.5

Multiply $2$ by $- 1$ .

$4 x^{2} - 2 x - 2 x - 1 \cdot - 1 - 8 = 0$

Step 2.1.3.1.6

Multiply $- 1$ by $- 1$ .

$4 x^{2} - 2 x - 2 x + 1 - 8 = 0$

Step 2.1.3.2

Subtract $2 x$ from $- 2 x$ .

$4 x^{2} - 4 x + 1 - 8 = 0$

Step 2.2

Subtract $8$ from $1$ .

$4 x^{2} - 4 x - 7 = 0$

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 $a = 4$ , $b = - 4$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

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

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

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

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

Step 5.1.2

Multiply $- 4 \cdot 4 \cdot - 7$ .

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

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

Step 5.1.2.2

Multiply $- 16$ by $- 7$ .

$x = \frac{4 \pm \sqrt{16 + 112}}{2 \cdot 4}$

Step 5.1.3

Add $16$ and $112$ .

$x = \frac{4 \pm \sqrt{128}}{2 \cdot 4}$

Step 5.1.4

Rewrite $128$ as $8^{2} \cdot 2$ .

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Step 5.1.4.1

Factor $64$ out of $128$ .

$x = \frac{4 \pm \sqrt{64 (2)}}{2 \cdot 4}$

Step 5.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{4 \pm \sqrt{8^{2} \cdot 2}}{2 \cdot 4}$

Step 5.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 8 \sqrt{2}}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{4 \pm 8 \sqrt{2}}{8}$

Step 5.3

Simplify $\frac{4 \pm 8 \sqrt{2}}{8}$ .

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

Step 6

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

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Step 6.1

Simplify the numerator.

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Step 6.1.1

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

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

Step 6.1.2

Multiply $- 4 \cdot 4 \cdot - 7$ .

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Step 6.1.2.1

Multiply $- 4$ by $4$ .

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

Step 6.1.2.2

Multiply $- 16$ by $- 7$ .

$x = \frac{4 \pm \sqrt{16 + 112}}{2 \cdot 4}$

Step 6.1.3

Add $16$ and $112$ .

$x = \frac{4 \pm \sqrt{128}}{2 \cdot 4}$

Step 6.1.4

Rewrite $128$ as $8^{2} \cdot 2$ .

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Step 6.1.4.1

Factor $64$ out of $128$ .

$x = \frac{4 \pm \sqrt{64 (2)}}{2 \cdot 4}$

Step 6.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{4 \pm \sqrt{8^{2} \cdot 2}}{2 \cdot 4}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 8 \sqrt{2}}{2 \cdot 4}$

Step 6.2

Multiply $2$ by $4$ .

$x = \frac{4 \pm 8 \sqrt{2}}{8}$

Step 6.3

Simplify $\frac{4 \pm 8 \sqrt{2}}{8}$ .

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

Step 6.4

Change the $\pm$ to $+$ .

$x = \frac{1 + 2 \sqrt{2}}{2}$

Step 7

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

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Step 7.1

Simplify the numerator.

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Step 7.1.1

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

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

Step 7.1.2

Multiply $- 4 \cdot 4 \cdot - 7$ .

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Step 7.1.2.1

Multiply $- 4$ by $4$ .

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

Step 7.1.2.2

Multiply $- 16$ by $- 7$ .

$x = \frac{4 \pm \sqrt{16 + 112}}{2 \cdot 4}$

Step 7.1.3

Add $16$ and $112$ .

$x = \frac{4 \pm \sqrt{128}}{2 \cdot 4}$

Step 7.1.4

Rewrite $128$ as $8^{2} \cdot 2$ .

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Step 7.1.4.1

Factor $64$ out of $128$ .

$x = \frac{4 \pm \sqrt{64 (2)}}{2 \cdot 4}$

Step 7.1.4.2

Rewrite $64$ as $8^{2}$ .

$x = \frac{4 \pm \sqrt{8^{2} \cdot 2}}{2 \cdot 4}$

Step 7.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 8 \sqrt{2}}{2 \cdot 4}$

Step 7.2

Multiply $2$ by $4$ .

$x = \frac{4 \pm 8 \sqrt{2}}{8}$

Step 7.3

Simplify $\frac{4 \pm 8 \sqrt{2}}{8}$ .

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

Step 7.4

Change the $\pm$ to $-$ .

$x = \frac{1 - 2 \sqrt{2}}{2}$

Step 8

The final answer is the combination of both solutions.

$x = \frac{1 + 2 \sqrt{2}}{2}, \frac{1 - 2 \sqrt{2}}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 + 2 \sqrt{2}}{2}, \frac{1 - 2 \sqrt{2}}{2}$

Decimal Form:

$x = 1.91421356 \dots, - 0.91421356 \dots$