Finite Math Examples

$- 12 x (7 x - 14) = 5$

Step 1

Divide each term in $- 12 x (7 x - 14) = 5$ by $- 12$ and simplify.

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Step 1.1

Divide each term in $- 12 x (7 x - 14) = 5$ by $- 12$ .

$\frac{- 12 x (7 x - 14)}{- 12} = \frac{5}{- 12}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify terms.

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Step 1.2.1.1

Cancel the common factor of $- 12$ .

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Step 1.2.1.1.1

Cancel the common factor.

$\frac{- 12 x (7 x - 14)}{- 12} = \frac{5}{- 12}$

Step 1.2.1.1.2

Divide $x (7 x - 14)$ by $1$ .

$x (7 x - 14) = \frac{5}{- 12}$

Step 1.2.1.2

Apply the distributive property.

$x (7 x) + x \cdot - 14 = \frac{5}{- 12}$

Step 1.2.1.3

Reorder.

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Step 1.2.1.3.1

Rewrite using the commutative property of multiplication.

$7 x \cdot x + x \cdot - 14 = \frac{5}{- 12}$

Step 1.2.1.3.2

Move $- 14$ to the left of $x$ .

$7 x \cdot x - 14 \cdot x = \frac{5}{- 12}$

Step 1.2.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.2.1

Move $x$ .

$7 (x \cdot x) - 14 \cdot x = \frac{5}{- 12}$

Step 1.2.2.2

Multiply $x$ by $x$ .

$7 x^{2} - 14 \cdot x = \frac{5}{- 12}$

$7 x^{2} - 14 x = \frac{5}{- 12}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$7 x^{2} - 14 x = - \frac{5}{12}$

Step 2

Add $\frac{5}{12}$ to both sides of the equation.

$7 x^{2} - 14 x + \frac{5}{12} = 0$

Step 3

Multiply through by the least common denominator $12$ , then simplify.

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Step 3.1

Apply the distributive property.

$12 (7 x^{2}) + 12 (- 14 x) + 12 (\frac{5}{12}) = 0$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $7$ by $12$ .

$84 x^{2} + 12 (- 14 x) + 12 (\frac{5}{12}) = 0$

Step 3.2.2

Multiply $- 14$ by $12$ .

$84 x^{2} - 168 x + 12 (\frac{5}{12}) = 0$

Step 3.2.3

Cancel the common factor of $12$ .

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Step 3.2.3.1

Cancel the common factor.

$84 x^{2} - 168 x + 12 (\frac{5}{12}) = 0$

Step 3.2.3.2

Rewrite the expression.

$84 x^{2} - 168 x + 5 = 0$

Step 4

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5

Substitute the values $a = 84$ , $b = - 168$ , and $c = 5$ into the quadratic formula and solve for $x$ .

$\frac{168 \pm \sqrt{{(- 168)}^{2} - 4 \cdot (84 \cdot 5)}}{2 \cdot 84}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $- 168$ to the power of $2$ .

$x = \frac{168 \pm \sqrt{28224 - 4 \cdot 84 \cdot 5}}{2 \cdot 84}$

Step 6.1.2

Multiply $- 4 \cdot 84 \cdot 5$ .

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Step 6.1.2.1

Multiply $- 4$ by $84$ .

$x = \frac{168 \pm \sqrt{28224 - 336 \cdot 5}}{2 \cdot 84}$

Step 6.1.2.2

Multiply $- 336$ by $5$ .

$x = \frac{168 \pm \sqrt{28224 - 1680}}{2 \cdot 84}$

Step 6.1.3

Subtract $1680$ from $28224$ .

$x = \frac{168 \pm \sqrt{26544}}{2 \cdot 84}$

Step 6.1.4

Rewrite $26544$ as $4^{2} \cdot 1659$ .

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Step 6.1.4.1

Factor $16$ out of $26544$ .

$x = \frac{168 \pm \sqrt{16 (1659)}}{2 \cdot 84}$

Step 6.1.4.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{168 \pm \sqrt{4^{2} \cdot 1659}}{2 \cdot 84}$

Step 6.1.5

Pull terms out from under the radical.

$x = \frac{168 \pm 4 \sqrt{1659}}{2 \cdot 84}$

Step 6.2

Multiply $2$ by $84$ .

$x = \frac{168 \pm 4 \sqrt{1659}}{168}$

Step 6.3

Simplify $\frac{168 \pm 4 \sqrt{1659}}{168}$ .

$x = \frac{42 \pm \sqrt{1659}}{42}$

Step 7

The final answer is the combination of both solutions.

$x = \frac{42 + \sqrt{1659}}{42}, \frac{42 - \sqrt{1659}}{42}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{42 + \sqrt{1659}}{42}, \frac{42 - \sqrt{1659}}{42}$

Decimal Form:

$x = 1.96978151 \dots, 0.03021848 \dots$