Pre-Algebra Examples

$12 x^{2} - 29 x = 14$

Step 1

Subtract $14$ from both sides of the equation.

$12 x^{2} - 29 x - 14 = 0$

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 $a = 12$ , $b = - 29$ , and $c = - 14$ into the quadratic formula and solve for $x$ .

$\frac{29 \pm \sqrt{{(- 29)}^{2} - 4 \cdot (12 \cdot - 14)}}{2 \cdot 12}$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $- 29$ to the power of $2$ .

$x = \frac{29 \pm \sqrt{841 - 4 \cdot 12 \cdot - 14}}{2 \cdot 12}$

Step 4.1.2

Multiply $- 4 \cdot 12 \cdot - 14$ .

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Step 4.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{29 \pm \sqrt{841 - 48 \cdot - 14}}{2 \cdot 12}$

Step 4.1.2.2

Multiply $- 48$ by $- 14$ .

$x = \frac{29 \pm \sqrt{841 + 672}}{2 \cdot 12}$

Step 4.1.3

Add $841$ and $672$ .

$x = \frac{29 \pm \sqrt{1513}}{2 \cdot 12}$

Step 4.2

Multiply $2$ by $12$ .

$x = \frac{29 \pm \sqrt{1513}}{24}$

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 $- 29$ to the power of $2$ .

$x = \frac{29 \pm \sqrt{841 - 4 \cdot 12 \cdot - 14}}{2 \cdot 12}$

Step 5.1.2

Multiply $- 4 \cdot 12 \cdot - 14$ .

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Step 5.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{29 \pm \sqrt{841 - 48 \cdot - 14}}{2 \cdot 12}$

Step 5.1.2.2

Multiply $- 48$ by $- 14$ .

$x = \frac{29 \pm \sqrt{841 + 672}}{2 \cdot 12}$

Step 5.1.3

Add $841$ and $672$ .

$x = \frac{29 \pm \sqrt{1513}}{2 \cdot 12}$

Step 5.2

Multiply $2$ by $12$ .

$x = \frac{29 \pm \sqrt{1513}}{24}$

Step 5.3

Change the $\pm$ to $+$ .

$x = \frac{29 + \sqrt{1513}}{24}$

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 $- 29$ to the power of $2$ .

$x = \frac{29 \pm \sqrt{841 - 4 \cdot 12 \cdot - 14}}{2 \cdot 12}$

Step 6.1.2

Multiply $- 4 \cdot 12 \cdot - 14$ .

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Step 6.1.2.1

Multiply $- 4$ by $12$ .

$x = \frac{29 \pm \sqrt{841 - 48 \cdot - 14}}{2 \cdot 12}$

Step 6.1.2.2

Multiply $- 48$ by $- 14$ .

$x = \frac{29 \pm \sqrt{841 + 672}}{2 \cdot 12}$

Step 6.1.3

Add $841$ and $672$ .

$x = \frac{29 \pm \sqrt{1513}}{2 \cdot 12}$

Step 6.2

Multiply $2$ by $12$ .

$x = \frac{29 \pm \sqrt{1513}}{24}$

Step 6.3

Change the $\pm$ to $-$ .

$x = \frac{29 - \sqrt{1513}}{24}$

Step 7

The final answer is the combination of both solutions.

$x = \frac{29 + \sqrt{1513}}{24}, \frac{29 - \sqrt{1513}}{24}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{29 + \sqrt{1513}}{24}, \frac{29 - \sqrt{1513}}{24}$

Decimal Form:

$x = 2.82905419 \dots, - 0.41238752 \dots$