Finite Math Examples

$24 \cdot (\frac{2 x}{3} \cdot \frac{x}{8}) = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1

Simplify $24 \cdot \frac{2 x}{3} \cdot \frac{x}{8}$ .

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Step 1.1

Rewrite.

$0 + 0 + 24 \cdot \frac{2 x}{3} \cdot \frac{x}{8} = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.2

Simplify by adding zeros.

$24 \cdot \frac{2 x}{3} \cdot \frac{x}{8} = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.3

Cancel the common factor of $8$ .

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Step 1.3.1

Factor $8$ out of $24 \cdot \frac{2 x}{3}$ .

$8 (3 \cdot \frac{2 x}{3}) \cdot \frac{x}{8} = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.3.2

Cancel the common factor.

$8 (3 \cdot \frac{2 x}{3}) \cdot \frac{x}{8} = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.3.3

Rewrite the expression.

$3 \cdot \frac{2 x}{3} \cdot x = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.4

Combine $3$ and $\frac{2 x}{3}$ .

$\frac{3 (2 x)}{3} \cdot x = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.5

Multiply $2$ by $3$ .

$\frac{6 x}{3} \cdot x = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.6

Combine $\frac{6 x}{3}$ and $x$ .

$\frac{6 x \cdot x}{3} = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.7

Raise $x$ to the power of $1$ .

$\frac{6 (x^{1} x)}{3} = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.8

Raise $x$ to the power of $1$ .

$\frac{6 (x^{1} x^{1})}{3} = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.9

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{6 x^{1 + 1}}{3} = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.10

Add $1$ and $1$ .

$\frac{6 x^{2}}{3} = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.11

Cancel the common factor of $6$ and $3$ .

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Step 1.11.1

Factor $3$ out of $6 x^{2}$ .

$\frac{3 (2 x^{2})}{3} = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.11.2

Cancel the common factors.

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Step 1.11.2.1

Factor $3$ out of $3$ .

$\frac{3 (2 x^{2})}{3 (1)} = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.11.2.2

Cancel the common factor.

$\frac{3 (2 x^{2})}{3 \cdot 1} = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.11.2.3

Rewrite the expression.

$\frac{2 x^{2}}{1} = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 1.11.2.4

Divide $2 x^{2}$ by $1$ .

$2 x^{2} = 24 \cdot (\frac{7}{2} + \frac{x}{4})$

Step 2

Simplify $24 \cdot (\frac{7}{2} + \frac{x}{4})$ .

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Step 2.1

Apply the distributive property.

$2 x^{2} = 24 (\frac{7}{2}) + 24 \frac{x}{4}$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Factor $2$ out of $24$ .

$2 x^{2} = 2 (12) \frac{7}{2} + 24 \frac{x}{4}$

Step 2.2.2

Cancel the common factor.

$2 x^{2} = 2 \cdot 12 \frac{7}{2} + 24 \frac{x}{4}$

Step 2.2.3

Rewrite the expression.

$2 x^{2} = 12 \cdot 7 + 24 \frac{x}{4}$

Step 2.3

Multiply $12$ by $7$ .

$2 x^{2} = 84 + 24 \frac{x}{4}$

Step 2.4

Cancel the common factor of $4$ .

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Step 2.4.1

Factor $4$ out of $24$ .

$2 x^{2} = 84 + 4 (6) \frac{x}{4}$

Step 2.4.2

Cancel the common factor.

$2 x^{2} = 84 + 4 \cdot 6 \frac{x}{4}$

Step 2.4.3

Rewrite the expression.

$2 x^{2} = 84 + 6 x$

Step 3

Subtract $6 x$ from both sides of the equation.

$2 x^{2} - 6 x = 84$

Step 4

Subtract $84$ from both sides of the equation.

$2 x^{2} - 6 x - 84 = 0$

Step 5

Factor $2$ out of $2 x^{2} - 6 x - 84$ .

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Step 5.1

Factor $2$ out of $2 x^{2}$ .

$2 (x^{2}) - 6 x - 84 = 0$

Step 5.2

Factor $2$ out of $- 6 x$ .

$2 (x^{2}) + 2 (- 3 x) - 84 = 0$

Step 5.3

Factor $2$ out of $- 84$ .

$2 x^{2} + 2 (- 3 x) + 2 \cdot - 42 = 0$

Step 5.4

Factor $2$ out of $2 x^{2} + 2 (- 3 x)$ .

$2 (x^{2} - 3 x) + 2 \cdot - 42 = 0$

Step 5.5

Factor $2$ out of $2 (x^{2} - 3 x) + 2 \cdot - 42$ .

$2 (x^{2} - 3 x - 42) = 0$

Step 6

Divide each term in $2 (x^{2} - 3 x - 42) = 0$ by $2$ and simplify.

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Step 6.1

Divide each term in $2 (x^{2} - 3 x - 42) = 0$ by $2$ .

$\frac{2 (x^{2} - 3 x - 42)}{2} = \frac{0}{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{2 (x^{2} - 3 x - 42)}{2} = \frac{0}{2}$

Step 6.2.1.2

Divide $x^{2} - 3 x - 42$ by $1$ .

$x^{2} - 3 x - 42 = \frac{0}{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $0$ by $2$ .

$x^{2} - 3 x - 42 = 0$

Step 7

Use the quadratic formula to find the solutions.

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

Step 8

Substitute the values $a = 1$ , $b = - 3$ , and $c = - 42$ into the quadratic formula and solve for $x$ .

$\frac{3 \pm \sqrt{{(- 3)}^{2} - 4 \cdot (1 \cdot - 42)}}{2 \cdot 1}$

Step 9

Simplify.

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Step 9.1

Simplify the numerator.

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Step 9.1.1

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 42}}{2 \cdot 1}$

Step 9.1.2

Multiply $- 4 \cdot 1 \cdot - 42$ .

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Step 9.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot - 42}}{2 \cdot 1}$

Step 9.1.2.2

Multiply $- 4$ by $- 42$ .

$x = \frac{3 \pm \sqrt{9 + 168}}{2 \cdot 1}$

Step 9.1.3

Add $9$ and $168$ .

$x = \frac{3 \pm \sqrt{177}}{2 \cdot 1}$

Step 9.2

Multiply $2$ by $1$ .

$x = \frac{3 \pm \sqrt{177}}{2}$

Step 10

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{177}}{2}, \frac{3 - \sqrt{177}}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{177}}{2}, \frac{3 - \sqrt{177}}{2}$

Decimal Form:

$x = 8.15206734 \dots, - 5.15206734 \dots$