Precalculus Examples

$f (- 8 x) = - 7 + \frac{6 (- 8 x) + 1}{- 8 x}$

Step 1

Factor each term.

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Step 1.1

Remove unnecessary parentheses.

$f \cdot - 8 x = - 7 + \frac{6 (- 8 x) + 1}{- 8 x}$

Step 1.2

Multiply $- 8$ by $6$ .

$f \cdot - 8 x = - 7 + \frac{- 48 x + 1}{- 8 x}$

Step 1.3

Move the negative in front of the fraction.

$f \cdot - 8 x = - 7 - \frac{- 48 x + 1}{8 x}$

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.

$1, 1, 1, 8 x$

Step 2.2

The LCM of one and any expression is the expression.

$8 x$

Step 3

Multiply each term in $f \cdot - 8 x = - 7 - \frac{- 48 x + 1}{8 x}$ by $8 x$ to eliminate the fractions.

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Step 3.1

Multiply each term in $f \cdot - 8 x = - 7 - \frac{- 48 x + 1}{8 x}$ by $8 x$ .

$f \cdot - 8 x (8 x) = - 7 (8 x) - \frac{- 48 x + 1}{8 x} (8 x)$

Step 3.2

Simplify the left side.

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Step 3.2.1

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

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Step 3.2.1.1

Move $x$ .

$f \cdot - 8 (x \cdot x) \cdot 8 = - 7 (8 x) - \frac{- 48 x + 1}{8 x} (8 x)$

Step 3.2.1.2

Multiply $x$ by $x$ .

$f \cdot - 8 x^{2} \cdot 8 = - 7 (8 x) - \frac{- 48 x + 1}{8 x} (8 x)$

Step 3.2.2

Move $- 8$ to the left of $f$ .

$- 8 \cdot f x^{2} \cdot 8 = - 7 (8 x) - \frac{- 48 x + 1}{8 x} (8 x)$

Step 3.2.3

Multiply $8$ by $- 8$ .

$- 64 f x^{2} = - 7 (8 x) - \frac{- 48 x + 1}{8 x} (8 x)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $8$ by $- 7$ .

$- 64 f x^{2} = - 56 x - \frac{- 48 x + 1}{8 x} (8 x)$

Step 3.3.1.2

Cancel the common factor of $8 x$ .

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Step 3.3.1.2.1

Move the leading negative in $- \frac{- 48 x + 1}{8 x}$ into the numerator.

$- 64 f x^{2} = - 56 x + \frac{- (- 48 x + 1)}{8 x} (8 x)$

Step 3.3.1.2.2

Cancel the common factor.

$- 64 f x^{2} = - 56 x + \frac{- (- 48 x + 1)}{8 x} (8 x)$

Step 3.3.1.2.3

Rewrite the expression.

$- 64 f x^{2} = - 56 x - (- 48 x + 1)$

Step 3.3.1.3

Apply the distributive property.

$- 64 f x^{2} = - 56 x - (- 48 x) - 1 \cdot 1$

Step 3.3.1.4

Multiply $- 48$ by $- 1$ .

$- 64 f x^{2} = - 56 x + 48 x - 1 \cdot 1$

Step 3.3.1.5

Multiply $- 1$ by $1$ .

$- 64 f x^{2} = - 56 x + 48 x - 1$

Step 3.3.2

Add $- 56 x$ and $48 x$ .

$- 64 f x^{2} = - 8 x - 1$

Step 4

Solve the equation.

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Step 4.1

Add $8 x$ to both sides of the equation.

$- 64 f x^{2} + 8 x = - 1$

Step 4.2

Add $1$ to both sides of the equation.

$- 64 f x^{2} + 8 x + 1 = 0$

Step 4.3

Use the quadratic formula to find the solutions.

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

Step 4.4

Substitute the values $a = - 64 f$ , $b = 8$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 8 \pm \sqrt{8^{2} - 4 \cdot (- 64 f \cdot 1)}}{2 (- 64 f)}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Raise $8$ to the power of $2$ .

$x = \frac{- 8 \pm \sqrt{64 - 4 \cdot (- 64 f) \cdot 1}}{2 (- 64 f)}$

Step 4.5.1.2

Multiply $- 4$ by $- 64$ .

$x = \frac{- 8 \pm \sqrt{64 + 256 f \cdot 1}}{2 (- 64 f)}$

Step 4.5.1.3

Multiply $256$ by $1$ .

$x = \frac{- 8 \pm \sqrt{64 + 256 f}}{2 (- 64 f)}$

Step 4.5.1.4

Factor $64$ out of $64 + 256 f$ .

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Step 4.5.1.4.1

Factor $64$ out of $64$ .

$x = \frac{- 8 \pm \sqrt{64 (1) + 256 f}}{2 (- 64 f)}$

Step 4.5.1.4.2

Factor $64$ out of $256 f$ .

$x = \frac{- 8 \pm \sqrt{64 (1) + 64 (4 f)}}{2 (- 64 f)}$

Step 4.5.1.4.3

Factor $64$ out of $64 (1) + 64 (4 f)$ .

$x = \frac{- 8 \pm \sqrt{64 (1 + 4 f)}}{2 (- 64 f)}$

Step 4.5.1.5

Rewrite $64 (1 + 4 f)$ as $8^{2} (1^{2} + 4 f)$ .

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Step 4.5.1.5.1

Rewrite $64$ as $8^{2}$ .

$x = \frac{- 8 \pm \sqrt{8^{2} (1 + 4 f)}}{2 (- 64 f)}$

Step 4.5.1.5.2

Rewrite $1$ as $1^{2}$ .

$x = \frac{- 8 \pm \sqrt{8^{2} (1^{2} + 4 f)}}{2 (- 64 f)}$

Step 4.5.1.6

Pull terms out from under the radical.

$x = \frac{- 8 \pm 8 \sqrt{1^{2} + 4 f}}{2 (- 64 f)}$

Step 4.5.1.7

One to any power is one.

$x = \frac{- 8 \pm 8 \sqrt{1 + 4 f}}{2 (- 64 f)}$

Step 4.5.2

Multiply $- 64$ by $2$ .

$x = \frac{- 8 \pm 8 \sqrt{1 + 4 f}}{- 128 f}$

Step 4.5.3

Simplify $\frac{- 8 \pm 8 \sqrt{1 + 4 f}}{- 128 f}$ .

$x = \frac{1 \pm \sqrt{1 + 4 f}}{16 f}$

Step 4.6

The final answer is the combination of both solutions.

$x = \frac{1 + \sqrt{1 + 4 f}}{16 f}$

$x = \frac{1 - \sqrt{1 + 4 f}}{16 f}$

$x = \frac{1 + \sqrt{1 + 4 f}}{16 f}$

$x = \frac{1 - \sqrt{1 + 4 f}}{16 f}$