Finite Math Examples

$\frac{x}{c - 2} = c x + 3$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

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

$c - 2, 1, 1$

Step 1.2

Remove parentheses.

$c - 2, 1, 1$

Step 1.3

The LCM of one and any expression is the expression.

$c - 2$

Step 2

Multiply each term in $\frac{x}{c - 2} = c x + 3$ by $c - 2$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{x}{c - 2} = c x + 3$ by $c - 2$ .

$\frac{x}{c - 2} (c - 2) = c x (c - 2) + 3 (c - 2)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $c - 2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{x}{c - 2} (c - 2) = c x (c - 2) + 3 (c - 2)$

Step 2.2.1.2

Rewrite the expression.

$x = c x (c - 2) + 3 (c - 2)$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Apply the distributive property.

$x = c x c + c x \cdot - 2 + 3 (c - 2)$

Step 2.3.1.2

Multiply $c$ by $c$ by adding the exponents.

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Step 2.3.1.2.1

Move $c$ .

$x = c \cdot c x + c x \cdot - 2 + 3 (c - 2)$

Step 2.3.1.2.2

Multiply $c$ by $c$ .

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

Step 2.3.1.3

Move $- 2$ to the left of $c x$ .

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

Step 2.3.1.4

Remove parentheses.

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

Step 2.3.1.5

Apply the distributive property.

$x = c^{2} x - 2 c x + 3 c + 3 \cdot - 2$

Step 2.3.1.6

Multiply $3$ by $- 2$ .

$x = c^{2} x - 2 c x + 3 c - 6$

Step 3

Solve the equation.

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Step 3.1

Rewrite the equation as $c^{2} x - 2 c x + 3 c - 6 = x$ .

$c^{2} x - 2 c x + 3 c - 6 = x$

Step 3.2

Subtract $x$ from both sides of the equation.

$c^{2} x - 2 c x + 3 c - 6 - x = 0$

Step 3.3

Use the quadratic formula to find the solutions.

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

Step 3.4

Substitute the values $a = x$ , $b = - 2 x + 3$ , and $c = - 6 - x$ into the quadratic formula and solve for $c$ .

$\frac{- (- 2 x + 3) \pm \sqrt{{(- 2 x + 3)}^{2} - 4 \cdot (x \cdot (- 6 - x))}}{2 x}$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Apply the distributive property.

$c = \frac{- (- 2 x) - 1 \cdot 3 \pm \sqrt{{(- 2 x + 3)}^{2} - 4 \cdot x \cdot (- 6 - x)}}{2 x}$

Step 3.5.2

Multiply $- 2$ by $- 1$ .

$c = \frac{2 x - 1 \cdot 3 \pm \sqrt{{(- 2 x + 3)}^{2} - 4 \cdot x \cdot (- 6 - x)}}{2 x}$

Step 3.5.3

Multiply $- 1$ by $3$ .

$c = \frac{2 x - 3 \pm \sqrt{{(- 2 x + 3)}^{2} - 4 \cdot x \cdot (- 6 - x)}}{2 x}$

Step 3.5.4

Rewrite ${(- 2 x + 3)}^{2}$ as $(- 2 x + 3) (- 2 x + 3)$ .

$c = \frac{2 x - 3 \pm \sqrt{(- 2 x + 3) (- 2 x + 3) - 4 \cdot x \cdot (- 6 - x)}}{2 x}$

Step 3.5.5

Expand $(- 2 x + 3) (- 2 x + 3)$ using the FOIL Method.

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Step 3.5.5.1

Apply the distributive property.

$c = \frac{2 x - 3 \pm \sqrt{- 2 x (- 2 x + 3) + 3 (- 2 x + 3) - 4 \cdot x \cdot (- 6 - x)}}{2 x}$

Step 3.5.5.2

Apply the distributive property.

$c = \frac{2 x - 3 \pm \sqrt{- 2 x (- 2 x) - 2 x \cdot 3 + 3 (- 2 x + 3) - 4 \cdot x \cdot (- 6 - x)}}{2 x}$

Step 3.5.5.3

Apply the distributive property.

$c = \frac{2 x - 3 \pm \sqrt{- 2 x (- 2 x) - 2 x \cdot 3 + 3 (- 2 x) + 3 \cdot 3 - 4 \cdot x \cdot (- 6 - x)}}{2 x}$

Step 3.5.6

Simplify and combine like terms.

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Step 3.5.6.1

Simplify each term.

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Step 3.5.6.1.1

Rewrite using the commutative property of multiplication.

$c = \frac{2 x - 3 \pm \sqrt{- 2 \cdot (- 2 x \cdot x) - 2 x \cdot 3 + 3 (- 2 x) + 3 \cdot 3 - 4 \cdot x \cdot (- 6 - x)}}{2 x}$

Step 3.5.6.1.2

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

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Step 3.5.6.1.2.1

Move $x$ .

$c = \frac{2 x - 3 \pm \sqrt{- 2 \cdot (- 2 (x \cdot x)) - 2 x \cdot 3 + 3 (- 2 x) + 3 \cdot 3 - 4 \cdot x \cdot (- 6 - x)}}{2 x}$

Step 3.5.6.1.2.2

Multiply $x$ by $x$ .

$c = \frac{2 x - 3 \pm \sqrt{- 2 \cdot (- 2 x^{2}) - 2 x \cdot 3 + 3 (- 2 x) + 3 \cdot 3 - 4 \cdot x \cdot (- 6 - x)}}{2 x}$

Step 3.5.6.1.3

Multiply $- 2$ by $- 2$ .

$c = \frac{2 x - 3 \pm \sqrt{4 x^{2} - 2 x \cdot 3 + 3 (- 2 x) + 3 \cdot 3 - 4 \cdot x \cdot (- 6 - x)}}{2 x}$

Step 3.5.6.1.4

Multiply $3$ by $- 2$ .

$c = \frac{2 x - 3 \pm \sqrt{4 x^{2} - 6 x + 3 (- 2 x) + 3 \cdot 3 - 4 \cdot x \cdot (- 6 - x)}}{2 x}$

Step 3.5.6.1.5

Multiply $- 2$ by $3$ .

$c = \frac{2 x - 3 \pm \sqrt{4 x^{2} - 6 x - 6 x + 3 \cdot 3 - 4 \cdot x \cdot (- 6 - x)}}{2 x}$

Step 3.5.6.1.6

Multiply $3$ by $3$ .

$c = \frac{2 x - 3 \pm \sqrt{4 x^{2} - 6 x - 6 x + 9 - 4 \cdot x \cdot (- 6 - x)}}{2 x}$

Step 3.5.6.2

Subtract $6 x$ from $- 6 x$ .

$c = \frac{2 x - 3 \pm \sqrt{4 x^{2} - 12 x + 9 - 4 \cdot x \cdot (- 6 - x)}}{2 x}$

Step 3.5.7

Apply the distributive property.

$c = \frac{2 x - 3 \pm \sqrt{4 x^{2} - 12 x + 9 - 4 x \cdot - 6 - 4 x (- x)}}{2 x}$

Step 3.5.8

Multiply $- 6$ by $- 4$ .

$c = \frac{2 x - 3 \pm \sqrt{4 x^{2} - 12 x + 9 + 24 x - 4 x (- x)}}{2 x}$

Step 3.5.9

Rewrite using the commutative property of multiplication.

$c = \frac{2 x - 3 \pm \sqrt{4 x^{2} - 12 x + 9 + 24 x - 4 \cdot (- 1 x \cdot x)}}{2 x}$

Step 3.5.10

Simplify each term.

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Step 3.5.10.1

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

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Step 3.5.10.1.1

Move $x$ .

$c = \frac{2 x - 3 \pm \sqrt{4 x^{2} - 12 x + 9 + 24 x - 4 \cdot (- 1 (x \cdot x))}}{2 x}$

Step 3.5.10.1.2

Multiply $x$ by $x$ .

$c = \frac{2 x - 3 \pm \sqrt{4 x^{2} - 12 x + 9 + 24 x - 4 \cdot (- 1 x^{2})}}{2 x}$

Step 3.5.10.2

Multiply $- 4$ by $- 1$ .

$c = \frac{2 x - 3 \pm \sqrt{4 x^{2} - 12 x + 9 + 24 x + 4 x^{2}}}{2 x}$

Step 3.5.11

Add $4 x^{2}$ and $4 x^{2}$ .

$c = \frac{2 x - 3 \pm \sqrt{8 x^{2} - 12 x + 9 + 24 x}}{2 x}$

Step 3.5.12

Add $- 12 x$ and $24 x$ .

$c = \frac{2 x - 3 \pm \sqrt{8 x^{2} + 12 x + 9}}{2 x}$

Step 3.6

The final answer is the combination of both solutions.

$c = \frac{2 x - 3 + \sqrt{8 x^{2} + 12 x + 9}}{2 x}$

$c = \frac{2 x - 3 - \sqrt{8 x^{2} + 12 x + 9}}{2 x}$

$c = \frac{2 x - 3 + \sqrt{8 x^{2} + 12 x + 9}}{2 x}$

$c = \frac{2 x - 3 - \sqrt{8 x^{2} + 12 x + 9}}{2 x}$