Pre-Algebra Examples

${(x - 6)}^{2} + 3 x = 29$

Step 1

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

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Step 1.1

Subtract $29$ from both sides of the equation.

${(x - 6)}^{2} + 3 x - 29 = 0$

Step 1.2

Simplify ${(x - 6)}^{2} + 3 x - 29$ .

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite ${(x - 6)}^{2}$ as $(x - 6) (x - 6)$ .

$(x - 6) (x - 6) + 3 x - 29 = 0$

Step 1.2.1.2

Expand $(x - 6) (x - 6)$ using the FOIL Method.

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Step 1.2.1.2.1

Apply the distributive property.

$x (x - 6) - 6 (x - 6) + 3 x - 29 = 0$

Step 1.2.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 6 - 6 (x - 6) + 3 x - 29 = 0$

Step 1.2.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 6 - 6 x - 6 \cdot - 6 + 3 x - 29 = 0$

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 $x$ .

$x^{2} + x \cdot - 6 - 6 x - 6 \cdot - 6 + 3 x - 29 = 0$

Step 1.2.1.3.1.2

Move $- 6$ to the left of $x$ .

$x^{2} - 6 \cdot x - 6 x - 6 \cdot - 6 + 3 x - 29 = 0$

Step 1.2.1.3.1.3

Multiply $- 6$ by $- 6$ .

$x^{2} - 6 x - 6 x + 36 + 3 x - 29 = 0$

Step 1.2.1.3.2

Subtract $6 x$ from $- 6 x$ .

$x^{2} - 12 x + 36 + 3 x - 29 = 0$

Step 1.2.2

Add $- 12 x$ and $3 x$ .

$x^{2} - 9 x + 36 - 29 = 0$

Step 1.2.3

Subtract $29$ from $36$ .

$x^{2} - 9 x + 7 = 0$

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 $a = 1$ , $b = - 9$ , and $c = 7$ into the quadratic formula and solve for $x$ .

$\frac{9 \pm \sqrt{{(- 9)}^{2} - 4 \cdot (1 \cdot 7)}}{2 \cdot 1}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Step 4.1.2

Multiply $- 4 \cdot 1 \cdot 7$ .

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Step 4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 7}}{2 \cdot 1}$

Step 4.1.2.2

Multiply $- 4$ by $7$ .

$x = \frac{9 \pm \sqrt{81 - 28}}{2 \cdot 1}$

Step 4.1.3

Subtract $28$ from $81$ .

$x = \frac{9 \pm \sqrt{53}}{2 \cdot 1}$

Step 4.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm \sqrt{53}}{2}$

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 $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Step 5.1.2

Multiply $- 4 \cdot 1 \cdot 7$ .

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Step 5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 7}}{2 \cdot 1}$

Step 5.1.2.2

Multiply $- 4$ by $7$ .

$x = \frac{9 \pm \sqrt{81 - 28}}{2 \cdot 1}$

Step 5.1.3

Subtract $28$ from $81$ .

$x = \frac{9 \pm \sqrt{53}}{2 \cdot 1}$

Step 5.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm \sqrt{53}}{2}$

Step 5.3

Change the $\pm$ to $+$ .

$x = \frac{9 + \sqrt{53}}{2}$

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 $- 9$ to the power of $2$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Step 6.1.2

Multiply $- 4 \cdot 1 \cdot 7$ .

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Step 6.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{9 \pm \sqrt{81 - 4 \cdot 7}}{2 \cdot 1}$

Step 6.1.2.2

Multiply $- 4$ by $7$ .

$x = \frac{9 \pm \sqrt{81 - 28}}{2 \cdot 1}$

Step 6.1.3

Subtract $28$ from $81$ .

$x = \frac{9 \pm \sqrt{53}}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$x = \frac{9 \pm \sqrt{53}}{2}$

Step 6.3

Change the $\pm$ to $-$ .

$x = \frac{9 - \sqrt{53}}{2}$

Step 7

The final answer is the combination of both solutions.

$x = \frac{9 + \sqrt{53}}{2}, \frac{9 - \sqrt{53}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9 + \sqrt{53}}{2}, \frac{9 - \sqrt{53}}{2}$

Decimal Form:

$x = 8.14005494 \dots, 0.85994505 \dots$