Finite Math Examples

\frac{- x^{2}}{{(1 + 4 x)}^{2}} = \frac{5}{4}

$\frac{- x^{2}}{{(1 + 4 x)}^{2}} = \frac{5}{4}$

Step 1

Move the negative in front of the fraction.

- \frac{x^{2}}{{(1 + 4 x)}^{2}} = \frac{5}{4}

$- \frac{x^{2}}{{(1 + 4 x)}^{2}} = \frac{5}{4}$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

{(1 + 4 x)}^{2}, 4

${(1 + 4 x)}^{2}, 4$

Step 2.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.3

The number

1

$1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.4

4

$4$ has factors of

2

$2$ and

2

$2$ .

2 \cdot 2

$2 \cdot 2$

Step 2.5

Multiply

2

$2$ by

2

$2$ .

4

$4$

Step 2.6

The factors for

1 + 4 x

$1 + 4 x$ are

(1 + 4 x) \cdot (1 + 4 x)

$(1 + 4 x) \cdot (1 + 4 x)$ , which is

1 + 4 x

$1 + 4 x$ multiplied by itself

2

$2$ times.

(1 + 4 x) = (1 + 4 x) \cdot (1 + 4 x)

$(1 + 4 x) = (1 + 4 x) \cdot (1 + 4 x)$

(1 + 4 x)

$(1 + 4 x)$ occurs

2

$2$ times.

Step 2.7

The LCM of

{(1 + 4 x)}^{2}

${(1 + 4 x)}^{2}$ is the result of multiplying all factors the greatest number of times they occur in either term.

{(1 + 4 x)}^{2}

${(1 + 4 x)}^{2}$

Step 2.8

The Least Common Multiple

LCM

$LCM$ of some numbers is the smallest number that the numbers are factors of.

4 {(1 + 4 x)}^{2}

$4 {(1 + 4 x)}^{2}$

4 {(1 + 4 x)}^{2}

$4 {(1 + 4 x)}^{2}$

Step 3

Multiply each term in

- \frac{x^{2}}{{(1 + 4 x)}^{2}} = \frac{5}{4}

$- \frac{x^{2}}{{(1 + 4 x)}^{2}} = \frac{5}{4}$ by

4 {(1 + 4 x)}^{2}

$4 {(1 + 4 x)}^{2}$ to eliminate the fractions.

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Step 3.1

Multiply each term in

- \frac{x^{2}}{{(1 + 4 x)}^{2}} = \frac{5}{4}

$- \frac{x^{2}}{{(1 + 4 x)}^{2}} = \frac{5}{4}$ by

4 {(1 + 4 x)}^{2}

$4 {(1 + 4 x)}^{2}$ .

- \frac{x^{2}}{{(1 + 4 x)}^{2}} (4 {(1 + 4 x)}^{2}) = \frac{5}{4} (4 {(1 + 4 x)}^{2})

$- \frac{x^{2}}{{(1 + 4 x)}^{2}} (4 {(1 + 4 x)}^{2}) = \frac{5}{4} (4 {(1 + 4 x)}^{2})$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of

{(1 + 4 x)}^{2}

${(1 + 4 x)}^{2}$ .

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Step 3.2.1.1

Move the leading negative in

- \frac{x^{2}}{{(1 + 4 x)}^{2}}

$- \frac{x^{2}}{{(1 + 4 x)}^{2}}$ into the numerator.

\frac{- x^{2}}{{(1 + 4 x)}^{2}} (4 {(1 + 4 x)}^{2}) = \frac{5}{4} (4 {(1 + 4 x)}^{2})

$\frac{- x^{2}}{{(1 + 4 x)}^{2}} (4 {(1 + 4 x)}^{2}) = \frac{5}{4} (4 {(1 + 4 x)}^{2})$

Step 3.2.1.2

Factor

{(1 + 4 x)}^{2}

${(1 + 4 x)}^{2}$ out of

4 {(1 + 4 x)}^{2}

$4 {(1 + 4 x)}^{2}$ .

\frac{- x^{2}}{{(1 + 4 x)}^{2}} ({(1 + 4 x)}^{2} \cdot 4) = \frac{5}{4} (4 {(1 + 4 x)}^{2})

$\frac{- x^{2}}{{(1 + 4 x)}^{2}} ({(1 + 4 x)}^{2} \cdot 4) = \frac{5}{4} (4 {(1 + 4 x)}^{2})$

Step 3.2.1.3

Cancel the common factor.

\frac{- x^{2}}{{(1 + 4 x)}^{2}} ({(1 + 4 x)}^{2} \cdot 4) = \frac{5}{4} (4 {(1 + 4 x)}^{2})

$\frac{- x^{2}}{{(1 + 4 x)}^{2}} ({(1 + 4 x)}^{2} \cdot 4) = \frac{5}{4} (4 {(1 + 4 x)}^{2})$

Step 3.2.1.4

Rewrite the expression.

- x^{2} \cdot 4 = \frac{5}{4} (4 {(1 + 4 x)}^{2})

$- x^{2} \cdot 4 = \frac{5}{4} (4 {(1 + 4 x)}^{2})$

- x^{2} \cdot 4 = \frac{5}{4} (4 {(1 + 4 x)}^{2})

$- x^{2} \cdot 4 = \frac{5}{4} (4 {(1 + 4 x)}^{2})$

Step 3.2.2

Multiply

4

$4$ by

- 1

$- 1$ .

- 4 x^{2} = \frac{5}{4} (4 {(1 + 4 x)}^{2})

$- 4 x^{2} = \frac{5}{4} (4 {(1 + 4 x)}^{2})$

- 4 x^{2} = \frac{5}{4} (4 {(1 + 4 x)}^{2})

$- 4 x^{2} = \frac{5}{4} (4 {(1 + 4 x)}^{2})$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of

4

$4$ .

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Step 3.3.1.1

Factor

4

$4$ out of

4 {(1 + 4 x)}^{2}

$4 {(1 + 4 x)}^{2}$ .

- 4 x^{2} = \frac{5}{4} (4 ({(1 + 4 x)}^{2}))

$- 4 x^{2} = \frac{5}{4} (4 ({(1 + 4 x)}^{2}))$

Step 3.3.1.2

Cancel the common factor.

- 4 x^{2} = \frac{5}{4} (4 {(1 + 4 x)}^{2})

$- 4 x^{2} = \frac{5}{4} (4 {(1 + 4 x)}^{2})$

Step 3.3.1.3

Rewrite the expression.

- 4 x^{2} = 5 {(1 + 4 x)}^{2}

$- 4 x^{2} = 5 {(1 + 4 x)}^{2}$

- 4 x^{2} = 5 {(1 + 4 x)}^{2}

$- 4 x^{2} = 5 {(1 + 4 x)}^{2}$

- 4 x^{2} = 5 {(1 + 4 x)}^{2}

$- 4 x^{2} = 5 {(1 + 4 x)}^{2}$

- 4 x^{2} = 5 {(1 + 4 x)}^{2}

$- 4 x^{2} = 5 {(1 + 4 x)}^{2}$

Step 4

Solve the equation.

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Step 4.1

Simplify

5 {(1 + 4 x)}^{2}

$5 {(1 + 4 x)}^{2}$ .

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Step 4.1.1

Rewrite

{(1 + 4 x)}^{2}

${(1 + 4 x)}^{2}$ as

(1 + 4 x) (1 + 4 x)

$(1 + 4 x) (1 + 4 x)$ .

- 4 x^{2} = 5 ((1 + 4 x) (1 + 4 x))

$- 4 x^{2} = 5 ((1 + 4 x) (1 + 4 x))$

Step 4.1.2

Expand

(1 + 4 x) (1 + 4 x)

$(1 + 4 x) (1 + 4 x)$ using the FOIL Method.

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Step 4.1.2.1

Apply the distributive property.

- 4 x^{2} = 5 (1 (1 + 4 x) + 4 x (1 + 4 x))

$- 4 x^{2} = 5 (1 (1 + 4 x) + 4 x (1 + 4 x))$

Step 4.1.2.2

Apply the distributive property.

- 4 x^{2} = 5 (1 \cdot 1 + 1 (4 x) + 4 x (1 + 4 x))

$- 4 x^{2} = 5 (1 \cdot 1 + 1 (4 x) + 4 x (1 + 4 x))$

Step 4.1.2.3

Apply the distributive property.

- 4 x^{2} = 5 (1 \cdot 1 + 1 (4 x) + 4 x \cdot 1 + 4 x (4 x))

$- 4 x^{2} = 5 (1 \cdot 1 + 1 (4 x) + 4 x \cdot 1 + 4 x (4 x))$

- 4 x^{2} = 5 (1 \cdot 1 + 1 (4 x) + 4 x \cdot 1 + 4 x (4 x))

$- 4 x^{2} = 5 (1 \cdot 1 + 1 (4 x) + 4 x \cdot 1 + 4 x (4 x))$

Step 4.1.3

Simplify and combine like terms.

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Step 4.1.3.1

Simplify each term.

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Step 4.1.3.1.1

Multiply

1

$1$ by

1

$1$ .

- 4 x^{2} = 5 (1 + 1 (4 x) + 4 x \cdot 1 + 4 x (4 x))

$- 4 x^{2} = 5 (1 + 1 (4 x) + 4 x \cdot 1 + 4 x (4 x))$

Step 4.1.3.1.2

Multiply

4 x

$4 x$ by

1

$1$ .

- 4 x^{2} = 5 (1 + 4 x + 4 x \cdot 1 + 4 x (4 x))

$- 4 x^{2} = 5 (1 + 4 x + 4 x \cdot 1 + 4 x (4 x))$

Step 4.1.3.1.3

Multiply

4

$4$ by

1

$1$ .

- 4 x^{2} = 5 (1 + 4 x + 4 x + 4 x (4 x))

$- 4 x^{2} = 5 (1 + 4 x + 4 x + 4 x (4 x))$

Step 4.1.3.1.4

Rewrite using the commutative property of multiplication.

- 4 x^{2} = 5 (1 + 4 x + 4 x + 4 \cdot 4 x \cdot x)

$- 4 x^{2} = 5 (1 + 4 x + 4 x + 4 \cdot 4 x \cdot x)$

Step 4.1.3.1.5

Multiply

x

$x$ by

x

$x$ by adding the exponents.

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Step 4.1.3.1.5.1

Move

x

$x$ .

- 4 x^{2} = 5 (1 + 4 x + 4 x + 4 \cdot 4 (x \cdot x))

$- 4 x^{2} = 5 (1 + 4 x + 4 x + 4 \cdot 4 (x \cdot x))$

Step 4.1.3.1.5.2

Multiply

x

$x$ by

x

$x$ .

- 4 x^{2} = 5 (1 + 4 x + 4 x + 4 \cdot 4 x^{2})

$- 4 x^{2} = 5 (1 + 4 x + 4 x + 4 \cdot 4 x^{2})$

- 4 x^{2} = 5 (1 + 4 x + 4 x + 4 \cdot 4 x^{2})

$- 4 x^{2} = 5 (1 + 4 x + 4 x + 4 \cdot 4 x^{2})$

Step 4.1.3.1.6

Multiply

4

$4$ by

4

$4$ .

- 4 x^{2} = 5 (1 + 4 x + 4 x + 16 x^{2})

$- 4 x^{2} = 5 (1 + 4 x + 4 x + 16 x^{2})$

- 4 x^{2} = 5 (1 + 4 x + 4 x + 16 x^{2})

$- 4 x^{2} = 5 (1 + 4 x + 4 x + 16 x^{2})$

Step 4.1.3.2

Add

4 x

$4 x$ and

4 x

$4 x$ .

- 4 x^{2} = 5 (1 + 8 x + 16 x^{2})

$- 4 x^{2} = 5 (1 + 8 x + 16 x^{2})$

- 4 x^{2} = 5 (1 + 8 x + 16 x^{2})

$- 4 x^{2} = 5 (1 + 8 x + 16 x^{2})$

Step 4.1.4

Apply the distributive property.

- 4 x^{2} = 5 \cdot 1 + 5 (8 x) + 5 (16 x^{2})

$- 4 x^{2} = 5 \cdot 1 + 5 (8 x) + 5 (16 x^{2})$

Step 4.1.5

Simplify.

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Step 4.1.5.1

Multiply

5

$5$ by

1

$1$ .

- 4 x^{2} = 5 + 5 (8 x) + 5 (16 x^{2})

$- 4 x^{2} = 5 + 5 (8 x) + 5 (16 x^{2})$

Step 4.1.5.2

Multiply

8

$8$ by

5

$5$ .

- 4 x^{2} = 5 + 40 x + 5 (16 x^{2})

$- 4 x^{2} = 5 + 40 x + 5 (16 x^{2})$

Step 4.1.5.3

Multiply

16

$16$ by

5

$5$ .

- 4 x^{2} = 5 + 40 x + 80 x^{2}

$- 4 x^{2} = 5 + 40 x + 80 x^{2}$

- 4 x^{2} = 5 + 40 x + 80 x^{2}

$- 4 x^{2} = 5 + 40 x + 80 x^{2}$

- 4 x^{2} = 5 + 40 x + 80 x^{2}

$- 4 x^{2} = 5 + 40 x + 80 x^{2}$

Step 4.2

Since

x

$x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

5 + 40 x + 80 x^{2} = - 4 x^{2}

$5 + 40 x + 80 x^{2} = - 4 x^{2}$

Step 4.3

Move all terms containing

x

$x$ to the left side of the equation.

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Step 4.3.1

Add

4 x^{2}

$4 x^{2}$ to both sides of the equation.

5 + 40 x + 80 x^{2} + 4 x^{2} = 0

$5 + 40 x + 80 x^{2} + 4 x^{2} = 0$

Step 4.3.2

Add

80 x^{2}

$80 x^{2}$ and

4 x^{2}

$4 x^{2}$ .

5 + 40 x + 84 x^{2} = 0

$5 + 40 x + 84 x^{2} = 0$

5 + 40 x + 84 x^{2} = 0

$5 + 40 x + 84 x^{2} = 0$

Step 4.4

Use the quadratic formula to find the solutions.

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

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

Step 4.5

Substitute the values

a = 84

$a = 84$ ,

b = 40

$b = 40$ , and

c = 5

$c = 5$ into the quadratic formula and solve for

x

$x$ .

\frac{- 40 \pm \sqrt{40^{2} - 4 \cdot (84 \cdot 5)}}{2 \cdot 84}

$\frac{- 40 \pm \sqrt{40^{2} - 4 \cdot (84 \cdot 5)}}{2 \cdot 84}$

Step 4.6

Simplify.

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Step 4.6.1

Simplify the numerator.

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Step 4.6.1.1

Raise

40

$40$ to the power of

2

$2$ .

x = \frac{- 40 \pm \sqrt{1600 - 4 \cdot 84 \cdot 5}}{2 \cdot 84}

$x = \frac{- 40 \pm \sqrt{1600 - 4 \cdot 84 \cdot 5}}{2 \cdot 84}$

Step 4.6.1.2

Multiply

- 4 \cdot 84 \cdot 5

$- 4 \cdot 84 \cdot 5$ .

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Step 4.6.1.2.1

Multiply

- 4

$- 4$ by

84

$84$ .

x = \frac{- 40 \pm \sqrt{1600 - 336 \cdot 5}}{2 \cdot 84}

$x = \frac{- 40 \pm \sqrt{1600 - 336 \cdot 5}}{2 \cdot 84}$

Step 4.6.1.2.2

Multiply

- 336

$- 336$ by

5

$5$ .

x = \frac{- 40 \pm \sqrt{1600 - 1680}}{2 \cdot 84}

$x = \frac{- 40 \pm \sqrt{1600 - 1680}}{2 \cdot 84}$

x = \frac{- 40 \pm \sqrt{1600 - 1680}}{2 \cdot 84}

$x = \frac{- 40 \pm \sqrt{1600 - 1680}}{2 \cdot 84}$

Step 4.6.1.3

Subtract

1680

$1680$ from

1600

$1600$ .

x = \frac{- 40 \pm \sqrt{- 80}}{2 \cdot 84}

$x = \frac{- 40 \pm \sqrt{- 80}}{2 \cdot 84}$

Step 4.6.1.4

Rewrite

- 80

$- 80$ as

- 1 (80)

$- 1 (80)$ .

x = \frac{- 40 \pm \sqrt{- 1 \cdot 80}}{2 \cdot 84}

$x = \frac{- 40 \pm \sqrt{- 1 \cdot 80}}{2 \cdot 84}$

Step 4.6.1.5

Rewrite

\sqrt{- 1 (80)}

$\sqrt{- 1 (80)}$ as

\sqrt{- 1} \cdot \sqrt{80}

$\sqrt{- 1} \cdot \sqrt{80}$ .

x = \frac{- 40 \pm \sqrt{- 1} \cdot \sqrt{80}}{2 \cdot 84}

$x = \frac{- 40 \pm \sqrt{- 1} \cdot \sqrt{80}}{2 \cdot 84}$

Step 4.6.1.6

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

x = \frac{- 40 \pm i \cdot \sqrt{80}}{2 \cdot 84}

$x = \frac{- 40 \pm i \cdot \sqrt{80}}{2 \cdot 84}$

Step 4.6.1.7

Rewrite

80

$80$ as

4^{2} \cdot 5

$4^{2} \cdot 5$ .

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Step 4.6.1.7.1

Factor

16

$16$ out of

80

$80$ .

x = \frac{- 40 \pm i \cdot \sqrt{16 (5)}}{2 \cdot 84}

$x = \frac{- 40 \pm i \cdot \sqrt{16 (5)}}{2 \cdot 84}$

Step 4.6.1.7.2

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

x = \frac{- 40 \pm i \cdot \sqrt{4^{2} \cdot 5}}{2 \cdot 84}

$x = \frac{- 40 \pm i \cdot \sqrt{4^{2} \cdot 5}}{2 \cdot 84}$

x = \frac{- 40 \pm i \cdot \sqrt{4^{2} \cdot 5}}{2 \cdot 84}

$x = \frac{- 40 \pm i \cdot \sqrt{4^{2} \cdot 5}}{2 \cdot 84}$

Step 4.6.1.8

Pull terms out from under the radical.

x = \frac{- 40 \pm i \cdot (4 \sqrt{5})}{2 \cdot 84}

$x = \frac{- 40 \pm i \cdot (4 \sqrt{5})}{2 \cdot 84}$

Step 4.6.1.9

Move

4

$4$ to the left of

i

$i$ .

x = \frac{- 40 \pm 4 i \sqrt{5}}{2 \cdot 84}

$x = \frac{- 40 \pm 4 i \sqrt{5}}{2 \cdot 84}$

x = \frac{- 40 \pm 4 i \sqrt{5}}{2 \cdot 84}

$x = \frac{- 40 \pm 4 i \sqrt{5}}{2 \cdot 84}$

Step 4.6.2

Multiply

2

$2$ by

84

$84$ .

x = \frac{- 40 \pm 4 i \sqrt{5}}{168}

$x = \frac{- 40 \pm 4 i \sqrt{5}}{168}$

Step 4.6.3

Simplify

\frac{- 40 \pm 4 i \sqrt{5}}{168}

$\frac{- 40 \pm 4 i \sqrt{5}}{168}$ .

x = \frac{- 10 \pm i \sqrt{5}}{42}

$x = \frac{- 10 \pm i \sqrt{5}}{42}$

x = \frac{- 10 \pm i \sqrt{5}}{42}

$x = \frac{- 10 \pm i \sqrt{5}}{42}$

Step 4.7

The final answer is the combination of both solutions.

x = - \frac{10 - i \sqrt{5}}{42}, - \frac{10 + i \sqrt{5}}{42}

$x = - \frac{10 - i \sqrt{5}}{42}, - \frac{10 + i \sqrt{5}}{42}$

x = \frac{- 10 \pm i \sqrt{5}}{42}

$x = \frac{- 10 \pm i \sqrt{5}}{42}$

Step 5