Finite Math Examples

$y = \frac{5}{x^{2} - 4 x + 7}$

Step 1

Rewrite the equation as $\frac{5}{x^{2} - 4 x + 7} = y$ .

$\frac{5}{x^{2} - 4 x + 7} = y$

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.

$x^{2} - 4 x + 7, 1$

Step 2.2

Remove parentheses.

$x^{2} - 4 x + 7, 1$

Step 2.3

The LCM of one and any expression is the expression.

$x^{2} - 4 x + 7$

Step 3

Multiply each term in $\frac{5}{x^{2} - 4 x + 7} = y$ by $x^{2} - 4 x + 7$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{5}{x^{2} - 4 x + 7} = y$ by $x^{2} - 4 x + 7$ .

$\frac{5}{x^{2} - 4 x + 7} (x^{2} - 4 x + 7) = y (x^{2} - 4 x + 7)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x^{2} - 4 x + 7$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{5}{x^{2} - 4 x + 7} (x^{2} - 4 x + 7) = y (x^{2} - 4 x + 7)$

Step 3.2.1.2

Rewrite the expression.

$5 = y (x^{2} - 4 x + 7)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Apply the distributive property.

$5 = y x^{2} + y (- 4 x) + y \cdot 7$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Rewrite using the commutative property of multiplication.

$5 = y x^{2} - 4 y x + y \cdot 7$

Step 3.3.2.2

Move $7$ to the left of $y$ .

$5 = y x^{2} - 4 y x + 7 \cdot y$

$5 = y x^{2} - 4 y x + 7 y$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $y x^{2} - 4 y x + 7 y = 5$ .

$y x^{2} - 4 y x + 7 y = 5$

Step 4.2

Subtract $5$ from both sides of the equation.

$y x^{2} - 4 y x + 7 y - 5 = 0$

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 $a = y$ , $b = - 4 y$ , and $c = 7 y - 5$ into the quadratic formula and solve for $x$ .

$\frac{4 y \pm \sqrt{{(- 4 y)}^{2} - 4 \cdot (y \cdot (7 y - 5))}}{2 y}$

Step 4.5

Simplify.

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Step 4.5.1

Simplify the numerator.

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Step 4.5.1.1

Add parentheses.

$x = \frac{4 y \pm \sqrt{{(- 4 y)}^{2} - 4 \cdot (y \cdot (7 y - 5))}}{2 y}$

Step 4.5.1.2

Let $u = y \cdot (7 y - 5)$ . Substitute $u$ for all occurrences of $y \cdot (7 y - 5)$ .

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Step 4.5.1.2.1

Apply the product rule to $- 4 y$ .

$x = \frac{4 y \pm \sqrt{{(- 4)}^{2} y^{2} - 4 \cdot u}}{2 y}$

Step 4.5.1.2.2

Raise $- 4$ to the power of $2$ .

$x = \frac{4 y \pm \sqrt{16 y^{2} - 4 u}}{2 y}$

Step 4.5.1.3

Factor $4$ out of $16 y^{2} - 4 u$ .

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Step 4.5.1.3.1

Factor $4$ out of $16 y^{2}$ .

$x = \frac{4 y \pm \sqrt{4 (4 y^{2}) - 4 u}}{2 y}$

Step 4.5.1.3.2

Factor $4$ out of $- 4 u$ .

$x = \frac{4 y \pm \sqrt{4 (4 y^{2}) + 4 (- u)}}{2 y}$

Step 4.5.1.3.3

Factor $4$ out of $4 (4 y^{2}) + 4 (- u)$ .

$x = \frac{4 y \pm \sqrt{4 (4 y^{2} - u)}}{2 y}$

Step 4.5.1.4

Replace all occurrences of $u$ with $y \cdot (7 y - 5)$ .

$x = \frac{4 y \pm \sqrt{4 (4 y^{2} - (y \cdot (7 y - 5)))}}{2 y}$

Step 4.5.1.5

Simplify.

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Step 4.5.1.5.1

Simplify each term.

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Step 4.5.1.5.1.1

Apply the distributive property.

$x = \frac{4 y \pm \sqrt{4 (4 y^{2} - (y (7 y) + y \cdot - 5))}}{2 y}$

Step 4.5.1.5.1.2

Rewrite using the commutative property of multiplication.

$x = \frac{4 y \pm \sqrt{4 (4 y^{2} - (7 y \cdot y + y \cdot - 5))}}{2 y}$

Step 4.5.1.5.1.3

Move $- 5$ to the left of $y$ .

$x = \frac{4 y \pm \sqrt{4 (4 y^{2} - (7 y \cdot y - 5 \cdot y))}}{2 y}$

Step 4.5.1.5.1.4

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

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Step 4.5.1.5.1.4.1

Move $y$ .

$x = \frac{4 y \pm \sqrt{4 (4 y^{2} - (7 (y \cdot y) - 5 \cdot y))}}{2 y}$

Step 4.5.1.5.1.4.2

Multiply $y$ by $y$ .

$x = \frac{4 y \pm \sqrt{4 (4 y^{2} - (7 y^{2} - 5 \cdot y))}}{2 y}$

$x = \frac{4 y \pm \sqrt{4 (4 y^{2} - (7 y^{2} - 5 y))}}{2 y}$

Step 4.5.1.5.1.5

Apply the distributive property.

$x = \frac{4 y \pm \sqrt{4 (4 y^{2} - (7 y^{2}) - (- 5 y))}}{2 y}$

Step 4.5.1.5.1.6

Multiply $7$ by $- 1$ .

$x = \frac{4 y \pm \sqrt{4 (4 y^{2} - 7 y^{2} - (- 5 y))}}{2 y}$

Step 4.5.1.5.1.7

Multiply $- 5$ by $- 1$ .

$x = \frac{4 y \pm \sqrt{4 (4 y^{2} - 7 y^{2} + 5 y)}}{2 y}$

Step 4.5.1.5.2

Subtract $7 y^{2}$ from $4 y^{2}$ .

$x = \frac{4 y \pm \sqrt{4 (- 3 y^{2} + 5 y)}}{2 y}$

Step 4.5.1.6

Factor $y$ out of $- 3 y^{2} + 5 y$ .

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Step 4.5.1.6.1

Factor $y$ out of $- 3 y^{2}$ .

$x = \frac{4 y \pm \sqrt{4 (y (- 3 y) + 5 y)}}{2 y}$

Step 4.5.1.6.2

Factor $y$ out of $5 y$ .

$x = \frac{4 y \pm \sqrt{4 (y (- 3 y) + y \cdot 5)}}{2 y}$

Step 4.5.1.6.3

Factor $y$ out of $y (- 3 y) + y \cdot 5$ .

$x = \frac{4 y \pm \sqrt{4 (y (- 3 y + 5))}}{2 y}$

$x = \frac{4 y \pm \sqrt{4 y (- 3 y + 5)}}{2 y}$

Step 4.5.1.7

Rewrite $4 y (- 3 y + 5)$ as $2^{2} (y (- 3 y + 5))$ .

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Step 4.5.1.7.1

Rewrite $4$ as $2^{2}$ .

$x = \frac{4 y \pm \sqrt{2^{2} y (- 3 y + 5)}}{2 y}$

Step 4.5.1.7.2

Add parentheses.

$x = \frac{4 y \pm \sqrt{2^{2} (y (- 3 y + 5))}}{2 y}$

Step 4.5.1.8

Pull terms out from under the radical.

$x = \frac{4 y \pm 2 \sqrt{y (- 3 y + 5)}}{2 y}$

Step 4.5.2

Simplify $\frac{4 y \pm 2 \sqrt{y (- 3 y + 5)}}{2 y}$ .

$x = \frac{2 y \pm \sqrt{y (- 3 y + 5)}}{y}$

Step 4.6

The final answer is the combination of both solutions.

$x = \frac{2 y + \sqrt{y (- 3 y + 5)}}{y}$

$x = \frac{2 y - \sqrt{y (- 3 y + 5)}}{y}$

$x = \frac{2 y + \sqrt{y (- 3 y + 5)}}{y}$

$x = \frac{2 y - \sqrt{y (- 3 y + 5)}}{y}$