Algebra Examples

${(x - 2)}^{2} - 20 = - 45$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Add $45$ to both sides of the equation.

${(x - 2)}^{2} - 20 + 45 = 0$

Step 1.2

Simplify ${(x - 2)}^{2} - 20 + 45$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

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

$(x - 2) (x - 2) - 20 + 45 = 0$

Step 1.2.1.2

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

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Step 1.2.1.2.1

Apply the distributive property.

$x (x - 2) - 2 (x - 2) - 20 + 45 = 0$

Step 1.2.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 (x - 2) - 20 + 45 = 0$

Step 1.2.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2 - 20 + 45 = 0$

Step 1.2.1.3

Simplify and combine like terms.

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Step 1.2.1.3.1

Simplify each term.

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Step 1.2.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2 - 20 + 45 = 0$

Step 1.2.1.3.1.2

Move $- 2$ to the left of $x$ .

$x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2 - 20 + 45 = 0$

Step 1.2.1.3.1.3

Multiply $- 2$ by $- 2$ .

$x^{2} - 2 x - 2 x + 4 - 20 + 45 = 0$

Step 1.2.1.3.2

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

$x^{2} - 4 x + 4 - 20 + 45 = 0$

Step 1.2.2

Subtract $20$ from $4$ .

$x^{2} - 4 x - 16 + 45 = 0$

Step 1.2.3

Add $- 16$ and $45$ .

$x^{2} - 4 x + 29 = 0$

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 $a = 1$ , $b = - 4$ , and $c = 29$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot 29)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot 29}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 29$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 29}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $29$ .

$x = \frac{4 \pm \sqrt{16 - 116}}{2 \cdot 1}$

Step 4.1.3

Subtract $116$ from $16$ .

$x = \frac{4 \pm \sqrt{- 100}}{2 \cdot 1}$

Step 4.1.4

Rewrite $- 100$ as $- 1 (100)$ .

$x = \frac{4 \pm \sqrt{- 1 \cdot 100}}{2 \cdot 1}$

Step 4.1.5

Rewrite $\sqrt{- 1 (100)}$ as $\sqrt{- 1} \cdot \sqrt{100}$ .

$x = \frac{4 \pm \sqrt{- 1} \cdot \sqrt{100}}{2 \cdot 1}$

Step 4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{4 \pm i \cdot \sqrt{100}}{2 \cdot 1}$

Step 4.1.7

Rewrite $100$ as $10^{2}$ .

$x = \frac{4 \pm i \cdot \sqrt{10^{2}}}{2 \cdot 1}$

Step 4.1.8

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

$x = \frac{4 \pm i \cdot 10}{2 \cdot 1}$

Step 4.1.9

Move $10$ to the left of $i$ .

$x = \frac{4 \pm 10 i}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 10 i}{2}$

Step 4.3

Simplify $\frac{4 \pm 10 i}{2}$ .

$x = 2 \pm 5 i$