Pre-Algebra Examples

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

Step 1

Move all terms containing $x$ to the left side of the equation.

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Step 1.1

Subtract $x$ from both sides of the equation.

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

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite ${(x - 8)}^{2}$ as $(x - 8) (x - 8)$ .

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

Step 1.2.2

Expand $(x - 8) (x - 8)$ using the FOIL Method.

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Step 1.2.2.1

Apply the distributive property.

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

Step 1.2.2.2

Apply the distributive property.

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

Step 1.2.2.3

Apply the distributive property.

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

Step 1.2.3

Simplify and combine like terms.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Multiply $x$ by $x$ .

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

Step 1.2.3.1.2

Move $- 8$ to the left of $x$ .

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

Step 1.2.3.1.3

Multiply $- 8$ by $- 8$ .

$x^{2} - 8 x - 8 x + 64 + {(x - 1)}^{2} - x = 0$

Step 1.2.3.2

Subtract $8 x$ from $- 8 x$ .

$x^{2} - 16 x + 64 + {(x - 1)}^{2} - x = 0$

Step 1.2.4

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

$x^{2} - 16 x + 64 + (x - 1) (x - 1) - x = 0$

Step 1.2.5

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

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Step 1.2.5.1

Apply the distributive property.

$x^{2} - 16 x + 64 + x (x - 1) - 1 (x - 1) - x = 0$

Step 1.2.5.2

Apply the distributive property.

$x^{2} - 16 x + 64 + x \cdot x + x \cdot - 1 - 1 (x - 1) - x = 0$

Step 1.2.5.3

Apply the distributive property.

$x^{2} - 16 x + 64 + x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 - x = 0$

Step 1.2.6

Simplify and combine like terms.

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Step 1.2.6.1

Simplify each term.

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Step 1.2.6.1.1

Multiply $x$ by $x$ .

$x^{2} - 16 x + 64 + x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1 - x = 0$

Step 1.2.6.1.2

Move $- 1$ to the left of $x$ .

$x^{2} - 16 x + 64 + x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1 - x = 0$

Step 1.2.6.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} - 16 x + 64 + x^{2} - x - 1 x - 1 \cdot - 1 - x = 0$

Step 1.2.6.1.4

Rewrite $- 1 x$ as $- x$ .

$x^{2} - 16 x + 64 + x^{2} - x - x - 1 \cdot - 1 - x = 0$

Step 1.2.6.1.5

Multiply $- 1$ by $- 1$ .

$x^{2} - 16 x + 64 + x^{2} - x - x + 1 - x = 0$

Step 1.2.6.2

Subtract $x$ from $- x$ .

$x^{2} - 16 x + 64 + x^{2} - 2 x + 1 - x = 0$

Step 1.3

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

$2 x^{2} - 16 x + 64 - 2 x + 1 - x = 0$

Step 1.4

Subtract $2 x$ from $- 16 x$ .

$2 x^{2} - 18 x + 64 + 1 - x = 0$

Step 1.5

Subtract $x$ from $- 18 x$ .

$2 x^{2} - 19 x + 64 + 1 = 0$

Step 1.6

Add $64$ and $1$ .

$2 x^{2} - 19 x + 65 = 0$

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 $a = 2$ , $b = - 19$ , and $c = 65$ into the quadratic formula and solve for $x$ .

$\frac{19 \pm \sqrt{{(- 19)}^{2} - 4 \cdot (2 \cdot 65)}}{2 \cdot 2}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 2 \cdot 65}}{2 \cdot 2}$

Step 4.1.2

Multiply $- 4 \cdot 2 \cdot 65$ .

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Step 4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{19 \pm \sqrt{361 - 8 \cdot 65}}{2 \cdot 2}$

Step 4.1.2.2

Multiply $- 8$ by $65$ .

$x = \frac{19 \pm \sqrt{361 - 520}}{2 \cdot 2}$

Step 4.1.3

Subtract $520$ from $361$ .

$x = \frac{19 \pm \sqrt{- 159}}{2 \cdot 2}$

Step 4.1.4

Rewrite $- 159$ as $- 1 (159)$ .

$x = \frac{19 \pm \sqrt{- 1 \cdot 159}}{2 \cdot 2}$

Step 4.1.5

Rewrite $\sqrt{- 1 (159)}$ as $\sqrt{- 1} \cdot \sqrt{159}$ .

$x = \frac{19 \pm \sqrt{- 1} \cdot \sqrt{159}}{2 \cdot 2}$

Step 4.1.6

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

$x = \frac{19 \pm i \sqrt{159}}{2 \cdot 2}$

Step 4.2

Multiply $2$ by $2$ .

$x = \frac{19 \pm i \sqrt{159}}{4}$

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 $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 2 \cdot 65}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot 65$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{19 \pm \sqrt{361 - 8 \cdot 65}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $65$ .

$x = \frac{19 \pm \sqrt{361 - 520}}{2 \cdot 2}$

Step 5.1.3

Subtract $520$ from $361$ .

$x = \frac{19 \pm \sqrt{- 159}}{2 \cdot 2}$

Step 5.1.4

Rewrite $- 159$ as $- 1 (159)$ .

$x = \frac{19 \pm \sqrt{- 1 \cdot 159}}{2 \cdot 2}$

Step 5.1.5

Rewrite $\sqrt{- 1 (159)}$ as $\sqrt{- 1} \cdot \sqrt{159}$ .

$x = \frac{19 \pm \sqrt{- 1} \cdot \sqrt{159}}{2 \cdot 2}$

Step 5.1.6

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

$x = \frac{19 \pm i \sqrt{159}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$x = \frac{19 \pm i \sqrt{159}}{4}$

Step 5.3

Change the $\pm$ to $+$ .

$x = \frac{19 + i \sqrt{159}}{4}$

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 $- 19$ to the power of $2$ .

$x = \frac{19 \pm \sqrt{361 - 4 \cdot 2 \cdot 65}}{2 \cdot 2}$

Step 6.1.2

Multiply $- 4 \cdot 2 \cdot 65$ .

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Step 6.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{19 \pm \sqrt{361 - 8 \cdot 65}}{2 \cdot 2}$

Step 6.1.2.2

Multiply $- 8$ by $65$ .

$x = \frac{19 \pm \sqrt{361 - 520}}{2 \cdot 2}$

Step 6.1.3

Subtract $520$ from $361$ .

$x = \frac{19 \pm \sqrt{- 159}}{2 \cdot 2}$

Step 6.1.4

Rewrite $- 159$ as $- 1 (159)$ .

$x = \frac{19 \pm \sqrt{- 1 \cdot 159}}{2 \cdot 2}$

Step 6.1.5

Rewrite $\sqrt{- 1 (159)}$ as $\sqrt{- 1} \cdot \sqrt{159}$ .

$x = \frac{19 \pm \sqrt{- 1} \cdot \sqrt{159}}{2 \cdot 2}$

Step 6.1.6

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

$x = \frac{19 \pm i \sqrt{159}}{2 \cdot 2}$

Step 6.2

Multiply $2$ by $2$ .

$x = \frac{19 \pm i \sqrt{159}}{4}$

Step 6.3

Change the $\pm$ to $-$ .

$x = \frac{19 - i \sqrt{159}}{4}$

Step 7

The final answer is the combination of both solutions.

$x = \frac{19 + i \sqrt{159}}{4}, \frac{19 - i \sqrt{159}}{4}$