Finite Math Examples

$y = x \div x^{2} + 5 x + 6$

Step 1

Rewrite the equation as $x \div x^{2} + 5 x + 6 = y$ .

$x \div x^{2} + 5 x + 6 = y$

Step 2

Reduce the expression $x \div x^{2}$ by cancelling the common factors.

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Step 2.1

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

$x^{1} \div x^{2} + 5 x + 6 = y$

Step 2.2

Factor $x$ out of $x^{1}$ .

$x \cdot 1 \div x^{2} + 5 x + 6 = y$

Step 2.3

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

$x \cdot 1 \div x \cdot x + 5 x + 6 = y$

Step 2.4

Cancel the common factor.

$x \cdot 1 \div x \cdot x + 5 x + 6 = y$

Step 2.5

Rewrite the expression.

$\frac{1}{x} + 5 x + 6 = y$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

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

$x, 1, 1, 1$

Step 3.2

The LCM of one and any expression is the expression.

$x$

Step 4

Multiply each term in $\frac{1}{x} + 5 x + 6 = y$ by $x$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{1}{x} + 5 x + 6 = y$ by $x$ .

$\frac{1}{x} x + 5 x \cdot x + 6 x = y x$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Cancel the common factor of $x$ .

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Step 4.2.1.1.1

Cancel the common factor.

$\frac{1}{x} x + 5 x \cdot x + 6 x = y x$

Step 4.2.1.1.2

Rewrite the expression.

$1 + 5 x \cdot x + 6 x = y x$

Step 4.2.1.2

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

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Step 4.2.1.2.1

Move $x$ .

$1 + 5 (x \cdot x) + 6 x = y x$

Step 4.2.1.2.2

Multiply $x$ by $x$ .

$1 + 5 x^{2} + 6 x = y x$

Step 5

Solve the equation.

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Step 5.1

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

$1 + 5 x^{2} + 6 x - y x = 0$

Step 5.2

Use the quadratic formula to find the solutions.

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

Step 5.3

Substitute the values $a = 5$ , $b = 6 - y$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- (6 - y) \pm \sqrt{{(6 - y)}^{2} - 4 \cdot (5 \cdot 1)}}{2 \cdot 5}$

Step 5.4

Simplify.

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Step 5.4.1

Simplify the numerator.

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Step 5.4.1.1

Apply the distributive property.

$x = \frac{- 1 \cdot 6 + y \pm \sqrt{{(6 - y)}^{2} - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.2

Multiply $- 1$ by $6$ .

$x = \frac{- 6 + y \pm \sqrt{{(6 - y)}^{2} - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.3

Multiply $- - y$ .

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Step 5.4.1.3.1

Multiply $- 1$ by $- 1$ .

$x = \frac{- 6 + 1 y \pm \sqrt{{(6 - y)}^{2} - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.3.2

Multiply $y$ by $1$ .

$x = \frac{- 6 + y \pm \sqrt{{(6 - y)}^{2} - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.4

Rewrite ${(6 - y)}^{2}$ as $(6 - y) (6 - y)$ .

$x = \frac{- 6 + y \pm \sqrt{(6 - y) (6 - y) - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.5

Expand $(6 - y) (6 - y)$ using the FOIL Method.

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Step 5.4.1.5.1

Apply the distributive property.

$x = \frac{- 6 + y \pm \sqrt{6 (6 - y) - y (6 - y) - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.5.2

Apply the distributive property.

$x = \frac{- 6 + y \pm \sqrt{6 \cdot 6 + 6 (- y) - y (6 - y) - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.5.3

Apply the distributive property.

$x = \frac{- 6 + y \pm \sqrt{6 \cdot 6 + 6 (- y) - y \cdot 6 - y (- y) - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.6

Simplify and combine like terms.

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Step 5.4.1.6.1

Simplify each term.

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Step 5.4.1.6.1.1

Multiply $6$ by $6$ .

$x = \frac{- 6 + y \pm \sqrt{36 + 6 (- y) - y \cdot 6 - y (- y) - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.6.1.2

Multiply $- 1$ by $6$ .

$x = \frac{- 6 + y \pm \sqrt{36 - 6 y - y \cdot 6 - y (- y) - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.6.1.3

Multiply $6$ by $- 1$ .

$x = \frac{- 6 + y \pm \sqrt{36 - 6 y - 6 y - y (- y) - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.6.1.4

Rewrite using the commutative property of multiplication.

$x = \frac{- 6 + y \pm \sqrt{36 - 6 y - 6 y - 1 \cdot (- 1 y \cdot y) - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.6.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 5.4.1.6.1.5.1

Move $y$ .

$x = \frac{- 6 + y \pm \sqrt{36 - 6 y - 6 y - 1 \cdot (- 1 (y \cdot y)) - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.6.1.5.2

Multiply $y$ by $y$ .

$x = \frac{- 6 + y \pm \sqrt{36 - 6 y - 6 y - 1 \cdot (- 1 y^{2}) - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.6.1.6

Multiply $- 1$ by $- 1$ .

$x = \frac{- 6 + y \pm \sqrt{36 - 6 y - 6 y + 1 y^{2} - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.6.1.7

Multiply $y^{2}$ by $1$ .

$x = \frac{- 6 + y \pm \sqrt{36 - 6 y - 6 y + y^{2} - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.6.2

Subtract $6 y$ from $- 6 y$ .

$x = \frac{- 6 + y \pm \sqrt{36 - 12 y + y^{2} - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.7

Multiply $- 4 \cdot 5 \cdot 1$ .

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Step 5.4.1.7.1

Multiply $- 4$ by $5$ .

$x = \frac{- 6 + y \pm \sqrt{36 - 12 y + y^{2} - 20 \cdot 1}}{2 \cdot 5}$

Step 5.4.1.7.2

Multiply $- 20$ by $1$ .

$x = \frac{- 6 + y \pm \sqrt{36 - 12 y + y^{2} - 20}}{2 \cdot 5}$

Step 5.4.1.8

Subtract $20$ from $36$ .

$x = \frac{- 6 + y \pm \sqrt{- 12 y + y^{2} + 16}}{2 \cdot 5}$

Step 5.4.1.9

Reorder terms.

$x = \frac{- 6 + y \pm \sqrt{y^{2} - 12 y + 16}}{2 \cdot 5}$

Step 5.4.2

Multiply $2$ by $5$ .

$x = \frac{- 6 + y \pm \sqrt{y^{2} - 12 y + 16}}{10}$

Step 5.5

The final answer is the combination of both solutions.

$x = - \frac{6 - y - \sqrt{y^{2} - 12 y + 16}}{10}$

$x = - \frac{6 - y + \sqrt{y^{2} - 12 y + 16}}{10}$

$x = - \frac{6 - y - \sqrt{y^{2} - 12 y + 16}}{10}$

$x = - \frac{6 - y + \sqrt{y^{2} - 12 y + 16}}{10}$