Finite Math Examples

$35 = \frac{2 x^{2} - 5 x + 2}{x^{2} + 8 x + 16}$

Step 1

Rewrite the equation as $\frac{2 x^{2} - 5 x + 2}{x^{2} + 8 x + 16} = 35$ .

$\frac{2 x^{2} - 5 x + 2}{x^{2} + 8 x + 16} = 35$

Step 2

Factor each term.

Tap for more steps...

Step 2.1

Factor by grouping.

Tap for more steps...

Step 2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = - 5$ .

Tap for more steps...

Step 2.1.1.1

Factor $- 5$ out of $- 5 x$ .

$\frac{2 x^{2} - 5 (x) + 2}{x^{2} + 8 x + 16} = 35$

Step 2.1.1.2

Rewrite $- 5$ as $- 1$ plus $- 4$

$\frac{2 x^{2} + (- 1 - 4) x + 2}{x^{2} + 8 x + 16} = 35$

Step 2.1.1.3

Apply the distributive property.

$\frac{2 x^{2} - 1 x - 4 x + 2}{x^{2} + 8 x + 16} = 35$

Step 2.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 2.1.2.1

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 1 x) - 4 x + 2}{x^{2} + 8 x + 16} = 35$

Step 2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (2 x - 1) - 2 (2 x - 1)}{x^{2} + 8 x + 16} = 35$

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$\frac{(2 x - 1) (x - 2)}{x^{2} + 8 x + 16} = 35$

Step 2.2

Factor using the perfect square rule.

Tap for more steps...

Step 2.2.1

Rewrite $16$ as $4^{2}$ .

$\frac{(2 x - 1) (x - 2)}{x^{2} + 8 x + 4^{2}} = 35$

Step 2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 x = 2 \cdot x \cdot 4$

Step 2.2.3

Rewrite the polynomial.

$\frac{(2 x - 1) (x - 2)}{x^{2} + 2 \cdot x \cdot 4 + 4^{2}} = 35$

Step 2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 4$ .

$\frac{(2 x - 1) (x - 2)}{{(x + 4)}^{2}} = 35$

Step 3

Find the LCD of the terms in the equation.

Tap for more steps...

Step 3.1

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

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

Step 3.2

The LCM of one and any expression is the expression.

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

Step 4

Multiply each term in $\frac{(2 x - 1) (x - 2)}{{(x + 4)}^{2}} = 35$ by ${(x + 4)}^{2}$ to eliminate the fractions.

Tap for more steps...

Step 4.1

Multiply each term in $\frac{(2 x - 1) (x - 2)}{{(x + 4)}^{2}} = 35$ by ${(x + 4)}^{2}$ .

$\frac{(2 x - 1) (x - 2)}{{(x + 4)}^{2}} {(x + 4)}^{2} = 35 {(x + 4)}^{2}$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of ${(x + 4)}^{2}$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$\frac{(2 x - 1) (x - 2)}{{(x + 4)}^{2}} {(x + 4)}^{2} = 35 {(x + 4)}^{2}$

Step 4.2.1.2

Rewrite the expression.

$(2 x - 1) (x - 2) = 35 {(x + 4)}^{2}$

Step 4.2.2

Expand $(2 x - 1) (x - 2)$ using the FOIL Method.

Tap for more steps...

Step 4.2.2.1

Apply the distributive property.

$2 x (x - 2) - 1 (x - 2) = 35 {(x + 4)}^{2}$

Step 4.2.2.2

Apply the distributive property.

$2 x \cdot x + 2 x \cdot - 2 - 1 (x - 2) = 35 {(x + 4)}^{2}$

Step 4.2.2.3

Apply the distributive property.

$2 x \cdot x + 2 x \cdot - 2 - 1 x - 1 \cdot - 2 = 35 {(x + 4)}^{2}$

Step 4.2.3

Simplify and combine like terms.

Tap for more steps...

Step 4.2.3.1

Simplify each term.

Tap for more steps...

Step 4.2.3.1.1

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

Tap for more steps...

Step 4.2.3.1.1.1

Move $x$ .

$2 (x \cdot x) + 2 x \cdot - 2 - 1 x - 1 \cdot - 2 = 35 {(x + 4)}^{2}$

Step 4.2.3.1.1.2

Multiply $x$ by $x$ .

$2 x^{2} + 2 x \cdot - 2 - 1 x - 1 \cdot - 2 = 35 {(x + 4)}^{2}$

Step 4.2.3.1.2

Multiply $- 2$ by $2$ .

$2 x^{2} - 4 x - 1 x - 1 \cdot - 2 = 35 {(x + 4)}^{2}$

Step 4.2.3.1.3

Rewrite $- 1 x$ as $- x$ .

$2 x^{2} - 4 x - x - 1 \cdot - 2 = 35 {(x + 4)}^{2}$

Step 4.2.3.1.4

Multiply $- 1$ by $- 2$ .

$2 x^{2} - 4 x - x + 2 = 35 {(x + 4)}^{2}$

Step 4.2.3.2

Subtract $x$ from $- 4 x$ .

$2 x^{2} - 5 x + 2 = 35 {(x + 4)}^{2}$

Step 5

Solve the equation.

Tap for more steps...

Step 5.1

Simplify $35 {(x + 4)}^{2}$ .

Tap for more steps...

Step 5.1.1

Rewrite.

$2 x^{2} - 5 x + 2 = 0 + 0 + 35 {(x + 4)}^{2}$

Step 5.1.2

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$2 x^{2} - 5 x + 2 = 35 ((x + 4) (x + 4))$

Step 5.1.3

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

Tap for more steps...

Step 5.1.3.1

Apply the distributive property.

$2 x^{2} - 5 x + 2 = 35 (x (x + 4) + 4 (x + 4))$

Step 5.1.3.2

Apply the distributive property.

$2 x^{2} - 5 x + 2 = 35 (x \cdot x + x \cdot 4 + 4 (x + 4))$

Step 5.1.3.3

Apply the distributive property.

$2 x^{2} - 5 x + 2 = 35 (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4)$

Step 5.1.4

Simplify and combine like terms.

Tap for more steps...

Step 5.1.4.1

Simplify each term.

Tap for more steps...

Step 5.1.4.1.1

Multiply $x$ by $x$ .

$2 x^{2} - 5 x + 2 = 35 (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4)$

Step 5.1.4.1.2

Move $4$ to the left of $x$ .

$2 x^{2} - 5 x + 2 = 35 (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4)$

Step 5.1.4.1.3

Multiply $4$ by $4$ .

$2 x^{2} - 5 x + 2 = 35 (x^{2} + 4 x + 4 x + 16)$

Step 5.1.4.2

Add $4 x$ and $4 x$ .

$2 x^{2} - 5 x + 2 = 35 (x^{2} + 8 x + 16)$

Step 5.1.5

Apply the distributive property.

$2 x^{2} - 5 x + 2 = 35 x^{2} + 35 (8 x) + 35 \cdot 16$

Step 5.1.6

Simplify.

Tap for more steps...

Step 5.1.6.1

Multiply $8$ by $35$ .

$2 x^{2} - 5 x + 2 = 35 x^{2} + 280 x + 35 \cdot 16$

Step 5.1.6.2

Multiply $35$ by $16$ .

$2 x^{2} - 5 x + 2 = 35 x^{2} + 280 x + 560$

Step 5.2

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 5.2.1

Subtract $35 x^{2}$ from both sides of the equation.

$2 x^{2} - 5 x + 2 - 35 x^{2} = 280 x + 560$

Step 5.2.2

Subtract $280 x$ from both sides of the equation.

$2 x^{2} - 5 x + 2 - 35 x^{2} - 280 x = 560$

Step 5.2.3

Subtract $35 x^{2}$ from $2 x^{2}$ .

$- 33 x^{2} - 5 x + 2 - 280 x = 560$

Step 5.2.4

Subtract $280 x$ from $- 5 x$ .

$- 33 x^{2} - 285 x + 2 = 560$

Step 5.3

Subtract $560$ from both sides of the equation.

$- 33 x^{2} - 285 x + 2 - 560 = 0$

Step 5.4

Subtract $560$ from $2$ .

$- 33 x^{2} - 285 x - 558 = 0$

Step 5.5

Factor the left side of the equation.

Tap for more steps...

Step 5.5.1

Factor $- 3$ out of $- 33 x^{2} - 285 x - 558$ .

Tap for more steps...

Step 5.5.1.1

Factor $- 3$ out of $- 33 x^{2}$ .

$- 3 (11 x^{2}) - 285 x - 558 = 0$

Step 5.5.1.2

Factor $- 3$ out of $- 285 x$ .

$- 3 (11 x^{2}) - 3 (95 x) - 558 = 0$

Step 5.5.1.3

Factor $- 3$ out of $- 558$ .

$- 3 (11 x^{2}) - 3 (95 x) - 3 \cdot 186 = 0$

Step 5.5.1.4

Factor $- 3$ out of $- 3 (11 x^{2}) - 3 (95 x)$ .

$- 3 (11 x^{2} + 95 x) - 3 \cdot 186 = 0$

Step 5.5.1.5

Factor $- 3$ out of $- 3 (11 x^{2} + 95 x) - 3 (186)$ .

$- 3 (11 x^{2} + 95 x + 186) = 0$

Step 5.5.2

Factor.

Tap for more steps...

Step 5.5.2.1

Factor by grouping.

Tap for more steps...

Step 5.5.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 11 \cdot 186 = 2046$ and whose sum is $b = 95$ .

Tap for more steps...

Step 5.5.2.1.1.1

Factor $95$ out of $95 x$ .

$- 3 (11 x^{2} + 95 (x) + 186) = 0$

Step 5.5.2.1.1.2

Rewrite $95$ as $33$ plus $62$

$- 3 (11 x^{2} + (33 + 62) x + 186) = 0$

Step 5.5.2.1.1.3

Apply the distributive property.

$- 3 (11 x^{2} + 33 x + 62 x + 186) = 0$

Step 5.5.2.1.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 5.5.2.1.2.1

Group the first two terms and the last two terms.

$- 3 ((11 x^{2} + 33 x) + 62 x + 186) = 0$

Step 5.5.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- 3 (11 x (x + 3) + 62 (x + 3)) = 0$

Step 5.5.2.1.3

Factor the polynomial by factoring out the greatest common factor, $x + 3$ .

$- 3 ((x + 3) (11 x + 62)) = 0$

Step 5.5.2.2

Remove unnecessary parentheses.

$- 3 (x + 3) (11 x + 62) = 0$

Step 5.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 3 = 0$

$11 x + 62 = 0$

Step 5.7

Set $x + 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.7.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 5.7.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 5.8

Set $11 x + 62$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 5.8.1

Set $11 x + 62$ equal to $0$ .

$11 x + 62 = 0$

Step 5.8.2

Solve $11 x + 62 = 0$ for $x$ .

Tap for more steps...

Step 5.8.2.1

Subtract $62$ from both sides of the equation.

$11 x = - 62$

Step 5.8.2.2

Divide each term in $11 x = - 62$ by $11$ and simplify.

Tap for more steps...

Step 5.8.2.2.1

Divide each term in $11 x = - 62$ by $11$ .

$\frac{11 x}{11} = \frac{- 62}{11}$

Step 5.8.2.2.2

Simplify the left side.

Tap for more steps...

Step 5.8.2.2.2.1

Cancel the common factor of $11$ .

Tap for more steps...

Step 5.8.2.2.2.1.1

Cancel the common factor.

$\frac{11 x}{11} = \frac{- 62}{11}$

Step 5.8.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 62}{11}$

Step 5.8.2.2.3

Simplify the right side.

Tap for more steps...

Step 5.8.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{62}{11}$

Step 5.9

The final solution is all the values that make $- 3 (x + 3) (11 x + 62) = 0$ true.

$x = - 3, - \frac{62}{11}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - 3, - \frac{62}{11}$

Decimal Form:

$x = - 3, - 5.\overline{63}$

Mixed Number Form:

$x = - 3, - 5 \frac{7}{11}$