Pre-Algebra Examples

$x^{2} + {(x - 5)}^{2} = 10$

Step 1

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

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Step 1.1

Subtract $10$ from both sides of the equation.

$x^{2} + {(x - 5)}^{2} - 10 = 0$

Step 1.2

Simplify $x^{2} + {(x - 5)}^{2} - 10$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$x^{2} + (x - 5) (x - 5) - 10 = 0$

Step 1.2.1.2

Expand $(x - 5) (x - 5)$ using the FOIL Method.

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Step 1.2.1.2.1

Apply the distributive property.

$x^{2} + x (x - 5) - 5 (x - 5) - 10 = 0$

Step 1.2.1.2.2

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot - 5 - 5 (x - 5) - 10 = 0$

Step 1.2.1.2.3

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 - 10 = 0$

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 $x$ .

$x^{2} + x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 - 10 = 0$

Step 1.2.1.3.1.2

Move $- 5$ to the left of $x$ .

$x^{2} + x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5 - 10 = 0$

Step 1.2.1.3.1.3

Multiply $- 5$ by $- 5$ .

$x^{2} + x^{2} - 5 x - 5 x + 25 - 10 = 0$

Step 1.2.1.3.2

Subtract $5 x$ from $- 5 x$ .

$x^{2} + x^{2} - 10 x + 25 - 10 = 0$

Step 1.2.2

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} - 10 x + 25 - 10 = 0$

Step 1.2.3

Subtract $10$ from $25$ .

$2 x^{2} - 10 x + 15 = 0$

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 $a = 2$ , $b = - 10$ , and $c = 15$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (2 \cdot 15)}}{2 \cdot 2}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 2 \cdot 15}}{2 \cdot 2}$

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot 15$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{10 \pm \sqrt{100 - 8 \cdot 15}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $15$ .

$x = \frac{10 \pm \sqrt{100 - 120}}{2 \cdot 2}$

Step 4.1.3

Subtract $120$ from $100$ .

$x = \frac{10 \pm \sqrt{- 20}}{2 \cdot 2}$

Step 4.1.4

Rewrite $- 20$ as $- 1 (20)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 20}}{2 \cdot 2}$

Step 4.1.5

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{20}}{2 \cdot 2}$

Step 4.1.6

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

$x = \frac{10 \pm i \cdot \sqrt{20}}{2 \cdot 2}$

Step 4.1.7

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 4.1.7.1

Factor $4$ out of $20$ .

$x = \frac{10 \pm i \cdot \sqrt{4 (5)}}{2 \cdot 2}$

Step 4.1.7.2

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

$x = \frac{10 \pm i \cdot \sqrt{2^{2} \cdot 5}}{2 \cdot 2}$

Step 4.1.8

Pull terms out from under the radical.

$x = \frac{10 \pm i \cdot (2 \sqrt{5})}{2 \cdot 2}$

Step 4.1.9

Move $2$ to the left of $i$ .

$x = \frac{10 \pm 2 i \sqrt{5}}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$x = \frac{10 \pm 2 i \sqrt{5}}{4}$

Step 4.3

Simplify $\frac{10 \pm 2 i \sqrt{5}}{4}$ .

$x = \frac{5 \pm i \sqrt{5}}{2}$

Step 5

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

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 2 \cdot 15}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot 15$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{10 \pm \sqrt{100 - 8 \cdot 15}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $15$ .

$x = \frac{10 \pm \sqrt{100 - 120}}{2 \cdot 2}$

Step 5.1.3

Subtract $120$ from $100$ .

$x = \frac{10 \pm \sqrt{- 20}}{2 \cdot 2}$

Step 5.1.4

Rewrite $- 20$ as $- 1 (20)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 20}}{2 \cdot 2}$

Step 5.1.5

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{20}}{2 \cdot 2}$

Step 5.1.6

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

$x = \frac{10 \pm i \cdot \sqrt{20}}{2 \cdot 2}$

Step 5.1.7

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 5.1.7.1

Factor $4$ out of $20$ .

$x = \frac{10 \pm i \cdot \sqrt{4 (5)}}{2 \cdot 2}$

Step 5.1.7.2

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

$x = \frac{10 \pm i \cdot \sqrt{2^{2} \cdot 5}}{2 \cdot 2}$

Step 5.1.8

Pull terms out from under the radical.

$x = \frac{10 \pm i \cdot (2 \sqrt{5})}{2 \cdot 2}$

Step 5.1.9

Move $2$ to the left of $i$ .

$x = \frac{10 \pm 2 i \sqrt{5}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$x = \frac{10 \pm 2 i \sqrt{5}}{4}$

Step 5.3

Simplify $\frac{10 \pm 2 i \sqrt{5}}{4}$ .

$x = \frac{5 \pm i \sqrt{5}}{2}$

Step 5.4

Change the $\pm$ to $+$ .

$x = \frac{5 + i \sqrt{5}}{2}$

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 $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 2 \cdot 15}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot 15$ .

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{10 \pm \sqrt{100 - 8 \cdot 15}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $15$ .

$x = \frac{10 \pm \sqrt{100 - 120}}{2 \cdot 2}$

Step 6.1.3

Subtract $120$ from $100$ .

$x = \frac{10 \pm \sqrt{- 20}}{2 \cdot 2}$

Step 6.1.4

Rewrite $- 20$ as $- 1 (20)$ .

$x = \frac{10 \pm \sqrt{- 1 \cdot 20}}{2 \cdot 2}$

Step 6.1.5

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$x = \frac{10 \pm \sqrt{- 1} \cdot \sqrt{20}}{2 \cdot 2}$

Step 6.1.6

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

$x = \frac{10 \pm i \cdot \sqrt{20}}{2 \cdot 2}$

Step 6.1.7

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 6.1.7.1

Factor $4$ out of $20$ .

$x = \frac{10 \pm i \cdot \sqrt{4 (5)}}{2 \cdot 2}$

Step 6.1.7.2

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

$x = \frac{10 \pm i \cdot \sqrt{2^{2} \cdot 5}}{2 \cdot 2}$

Step 6.1.8

Pull terms out from under the radical.

$x = \frac{10 \pm i \cdot (2 \sqrt{5})}{2 \cdot 2}$

Step 6.1.9

Move $2$ to the left of $i$ .

$x = \frac{10 \pm 2 i \sqrt{5}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$x = \frac{10 \pm 2 i \sqrt{5}}{4}$

Step 6.3

Simplify $\frac{10 \pm 2 i \sqrt{5}}{4}$ .

$x = \frac{5 \pm i \sqrt{5}}{2}$

Step 6.4

Change the $\pm$ to $-$ .

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

Step 7

The final answer is the combination of both solutions.

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