Finite Math Examples

$8 x \cdot 1 + 9 x \cdot 2 = 117$ , $4 x \cdot 1 + 6 x \cdot 2 = 66$

Step 1

Simplify $8 x \cdot 1 + 9 x \cdot 2$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Multiply $8$ by $1$ .

$8 x + 9 x \cdot 2 = 117$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

Step 1.1.2

Multiply $2$ by $9$ .

$8 x + 18 x = 117$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

$8 x + 18 x = 117$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

Step 1.2

Add $8 x$ and $18 x$ .

$26 x = 117$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

$26 x = 117$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

Step 2

Divide each term in $26 x = 117$ by $26$ and simplify.

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Step 2.1

Divide each term in $26 x = 117$ by $26$ .

$\frac{26 x}{26} = \frac{117}{26}$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $26$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{26 x}{26} = \frac{117}{26}$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

Step 2.2.1.2

Divide $x$ by $1$ .

$x = \frac{117}{26}$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

$x = \frac{117}{26}$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

$x = \frac{117}{26}$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $117$ and $26$ .

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Step 2.3.1.1

Factor $13$ out of $117$ .

$x = \frac{13 (9)}{26}$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $13$ out of $26$ .

$x = \frac{13 \cdot 9}{13 \cdot 2}$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

Step 2.3.1.2.2

Cancel the common factor.

$x = \frac{13 \cdot 9}{13 \cdot 2}$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

Step 2.3.1.2.3

Rewrite the expression.

$x = \frac{9}{2}$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

$x = \frac{9}{2}$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

$x = \frac{9}{2}$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

$x = \frac{9}{2}$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

$x = \frac{9}{2}$

$4 x \cdot 1 + 6 x \cdot 2 = 66$

Step 3

Simplify $4 x \cdot 1 + 6 x \cdot 2$ .

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $4$ by $1$ .

$4 x + 6 x \cdot 2 = 66$

$x = \frac{9}{2}$

Step 3.1.2

Multiply $2$ by $6$ .

$4 x + 12 x = 66$

$x = \frac{9}{2}$

$4 x + 12 x = 66$

$x = \frac{9}{2}$

Step 3.2

Add $4 x$ and $12 x$ .

$16 x = 66$

$x = \frac{9}{2}$

$16 x = 66$

$x = \frac{9}{2}$

Step 4

Divide each term in $16 x = 66$ by $16$ and simplify.

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Step 4.1

Divide each term in $16 x = 66$ by $16$ .

$\frac{16 x}{16} = \frac{66}{16}$

$x = \frac{9}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $16$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{16 x}{16} = \frac{66}{16}$

$x = \frac{9}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{66}{16}$

$x = \frac{9}{2}$

$x = \frac{66}{16}$

$x = \frac{9}{2}$

$x = \frac{66}{16}$

$x = \frac{9}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $66$ and $16$ .

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Step 4.3.1.1

Factor $2$ out of $66$ .

$x = \frac{2 (33)}{16}$

$x = \frac{9}{2}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $2$ out of $16$ .

$x = \frac{2 \cdot 33}{2 \cdot 8}$

$x = \frac{9}{2}$

Step 4.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 33}{2 \cdot 8}$

$x = \frac{9}{2}$

Step 4.3.1.2.3

Rewrite the expression.

$x = \frac{33}{8}$

$x = \frac{9}{2}$

$x = \frac{33}{8}$

$x = \frac{9}{2}$

$x = \frac{33}{8}$

$x = \frac{9}{2}$

$x = \frac{33}{8}$

$x = \frac{9}{2}$

$x = \frac{33}{8}$

$x = \frac{9}{2}$

Step 5

Since the roots of an equation are the points where the solution is $0$ , set each root as a factor of the equation that equals $0$ .

$(x - \frac{9}{2}) (x - \frac{33}{8}) = 0$

Step 6

Simplify.

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Step 6.1

Expand $(x - \frac{9}{2}) (x - \frac{33}{8})$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$x (x - \frac{33}{8}) - \frac{9}{2} \cdot (x - \frac{33}{8}) = 0$

Step 6.1.2

Apply the distributive property.

$x \cdot x + x (- \frac{33}{8}) - \frac{9}{2} \cdot (x - \frac{33}{8}) = 0$

Step 6.1.3

Apply the distributive property.

$x \cdot x + x (- \frac{33}{8}) - \frac{9}{2} x - \frac{9}{2} \cdot (- \frac{33}{8}) = 0$

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

$x^{2} + x (- \frac{33}{8}) - \frac{9}{2} x - \frac{9}{2} \cdot (- \frac{33}{8}) = 0$

Step 6.2.1.2

Combine $x$ and $\frac{33}{8}$ .

$x^{2} - \frac{x \cdot 33}{8} - \frac{9}{2} x - \frac{9}{2} \cdot (- \frac{33}{8}) = 0$

Step 6.2.1.3

Move $33$ to the left of $x$ .

$x^{2} - \frac{33 \cdot x}{8} - \frac{9}{2} x - \frac{9}{2} \cdot (- \frac{33}{8}) = 0$

Step 6.2.1.4

Combine $x$ and $\frac{9}{2}$ .

$x^{2} - \frac{33 x}{8} - \frac{x \cdot 9}{2} - \frac{9}{2} \cdot (- \frac{33}{8}) = 0$

Step 6.2.1.5

Move $9$ to the left of $x$ .

$x^{2} - \frac{33 x}{8} - \frac{9 \cdot x}{2} - \frac{9}{2} \cdot (- \frac{33}{8}) = 0$

Step 6.2.1.6

Multiply $- \frac{9}{2} (- \frac{33}{8})$ .

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Step 6.2.1.6.1

Multiply $- 1$ by $- 1$ .

$x^{2} - \frac{33 x}{8} - \frac{9 x}{2} + 1 (\frac{9}{2}) \cdot \frac{33}{8} = 0$

Step 6.2.1.6.2

Multiply $\frac{9}{2}$ by $1$ .

$x^{2} - \frac{33 x}{8} - \frac{9 x}{2} + \frac{9}{2} \cdot \frac{33}{8} = 0$

Step 6.2.1.6.3

Multiply $\frac{9}{2}$ by $\frac{33}{8}$ .

$x^{2} - \frac{33 x}{8} - \frac{9 x}{2} + \frac{9 \cdot 33}{2 \cdot 8} = 0$

Step 6.2.1.6.4

Multiply $9$ by $33$ .

$x^{2} - \frac{33 x}{8} - \frac{9 x}{2} + \frac{297}{2 \cdot 8} = 0$

Step 6.2.1.6.5

Multiply $2$ by $8$ .

$x^{2} - \frac{33 x}{8} - \frac{9 x}{2} + \frac{297}{16} = 0$

Step 6.2.2

To write $- \frac{9 x}{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} - \frac{33 x}{8} - \frac{9 x}{2} \cdot \frac{4}{4} + \frac{297}{16} = 0$

Step 6.2.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.3.1

Multiply $\frac{9 x}{2}$ by $\frac{4}{4}$ .

$x^{2} - \frac{33 x}{8} - \frac{9 x \cdot 4}{2 \cdot 4} + \frac{297}{16} = 0$

Step 6.2.3.2

Multiply $2$ by $4$ .

$x^{2} - \frac{33 x}{8} - \frac{9 x \cdot 4}{8} + \frac{297}{16} = 0$

Step 6.2.4

Combine the numerators over the common denominator.

$x^{2} + \frac{- 33 x - 9 x \cdot 4}{8} + \frac{297}{16} = 0$

Step 6.2.5

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$x^{2} \cdot \frac{8}{8} + \frac{- 33 x - 9 x \cdot 4}{8} + \frac{297}{16} = 0$

Step 6.2.6

Combine $x^{2}$ and $\frac{8}{8}$ .

$\frac{x^{2} \cdot 8}{8} + \frac{- 33 x - 9 x \cdot 4}{8} + \frac{297}{16} = 0$

Step 6.2.7

Combine the numerators over the common denominator.

$\frac{x^{2} \cdot 8 - 33 x - 9 x \cdot 4}{8} + \frac{297}{16} = 0$

Step 6.2.8

To write $\frac{x^{2} \cdot 8 - 33 x - 9 x \cdot 4}{8}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x^{2} \cdot 8 - 33 x - 9 x \cdot 4}{8} \cdot \frac{2}{2} + \frac{297}{16} = 0$

Step 6.2.9

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.9.1

Multiply $\frac{x^{2} \cdot 8 - 33 x - 9 x \cdot 4}{8}$ by $\frac{2}{2}$ .

$\frac{(x^{2} \cdot 8 - 33 x - 9 x \cdot 4) \cdot 2}{8 \cdot 2} + \frac{297}{16} = 0$

Step 6.2.9.2

Multiply $8$ by $2$ .

$\frac{(x^{2} \cdot 8 - 33 x - 9 x \cdot 4) \cdot 2}{16} + \frac{297}{16} = 0$

Step 6.2.10

Combine the numerators over the common denominator.

$\frac{(x^{2} \cdot 8 - 33 x - 9 x \cdot 4) \cdot 2 + 297}{16} = 0$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Move $8$ to the left of $x^{2}$ .

$\frac{(8 \cdot x^{2} - 33 x - 9 x \cdot 4) \cdot 2 + 297}{16} = 0$

Step 6.3.1.2

Multiply $4$ by $- 9$ .

$\frac{(8 x^{2} - 33 x - 36 x) \cdot 2 + 297}{16} = 0$

Step 6.3.2

Subtract $36 x$ from $- 33 x$ .

$\frac{(8 x^{2} - 69 x) \cdot 2 + 297}{16} = 0$

Step 6.3.3

Apply the distributive property.

$\frac{8 x^{2} \cdot 2 - 69 x \cdot 2 + 297}{16} = 0$

Step 6.3.4

Multiply $2$ by $8$ .

$\frac{16 x^{2} - 69 x \cdot 2 + 297}{16} = 0$

Step 6.3.5

Multiply $2$ by $- 69$ .

$\frac{16 x^{2} - 138 x + 297}{16} = 0$

Step 6.3.6

Factor by grouping.

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Step 6.3.6.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 16 \cdot 297 = 4752$ and whose sum is $b = - 138$ .

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Step 6.3.6.1.1

Factor $- 138$ out of $- 138 x$ .

$\frac{16 x^{2} - 138 x + 297}{16} = 0$

Step 6.3.6.1.2

Rewrite $- 138$ as $- 66$ plus $- 72$

$\frac{16 x^{2} + (- 66 - 72) x + 297}{16} = 0$

Step 6.3.6.1.3

Apply the distributive property.

$\frac{16 x^{2} - 66 x - 72 x + 297}{16} = 0$

Step 6.3.6.2

Factor out the greatest common factor from each group.

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Step 6.3.6.2.1

Group the first two terms and the last two terms.

$\frac{(16 x^{2} - 66 x) - 72 x + 297}{16} = 0$

Step 6.3.6.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{2 x (8 x - 33) - 9 (8 x - 33)}{16} = 0$

Step 6.3.6.3

Factor the polynomial by factoring out the greatest common factor, $8 x - 33$ .

$\frac{(8 x - 33) (2 x - 9)}{16} = 0$