Algebra Examples

$\frac{3 x - 12}{3 x} = \frac{4 x^{2} - 9}{4 x^{2} - 16 x + 15}$

Step 1

Simplify both sides.

Tap for more steps...

Step 1.1

Cancel the common factor of $3 x - 12$ and $3$ .

Tap for more steps...

Step 1.1.1

Factor $3$ out of $3 x$ .

$\frac{3 (x) - 12}{3 x} = \frac{4 x^{2} - 9}{4 x^{2} - 16 x + 15}$

Step 1.1.2

Factor $3$ out of $- 12$ .

$\frac{3 (x) + 3 \cdot - 4}{3 x} = \frac{4 x^{2} - 9}{4 x^{2} - 16 x + 15}$

Step 1.1.3

Factor $3$ out of $3 (x) + 3 (- 4)$ .

$\frac{3 (x - 4)}{3 x} = \frac{4 x^{2} - 9}{4 x^{2} - 16 x + 15}$

Step 1.1.4

Cancel the common factors.

Tap for more steps...

Step 1.1.4.1

Factor $3$ out of $3 x$ .

$\frac{3 (x - 4)}{3 (x)} = \frac{4 x^{2} - 9}{4 x^{2} - 16 x + 15}$

Step 1.1.4.2

Cancel the common factor.

$\frac{3 (x - 4)}{3 x} = \frac{4 x^{2} - 9}{4 x^{2} - 16 x + 15}$

Step 1.1.4.3

Rewrite the expression.

$\frac{x - 4}{x} = \frac{4 x^{2} - 9}{4 x^{2} - 16 x + 15}$

Step 1.2

Simplify the numerator.

Tap for more steps...

Step 1.2.1

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

$\frac{x - 4}{x} = \frac{{(2 x)}^{2} - 9}{4 x^{2} - 16 x + 15}$

Step 1.2.2

Rewrite $9$ as $3^{2}$ .

$\frac{x - 4}{x} = \frac{{(2 x)}^{2} - 3^{2}}{4 x^{2} - 16 x + 15}$

Step 1.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 3$ .

$\frac{x - 4}{x} = \frac{(2 x + 3) (2 x - 3)}{4 x^{2} - 16 x + 15}$

Step 1.3

Factor by grouping.

Tap for more steps...

Step 1.3.1

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 $a \cdot c = 4 \cdot 15 = 60$ and whose sum is $b = - 16$ .

Tap for more steps...

Step 1.3.1.1

Factor $- 16$ out of $- 16 x$ .

$\frac{x - 4}{x} = \frac{(2 x + 3) (2 x - 3)}{4 x^{2} - 16 (x) + 15}$

Step 1.3.1.2

Rewrite $- 16$ as $- 6$ plus $- 10$

$\frac{x - 4}{x} = \frac{(2 x + 3) (2 x - 3)}{4 x^{2} + (- 6 - 10) x + 15}$

Step 1.3.1.3

Apply the distributive property.

$\frac{x - 4}{x} = \frac{(2 x + 3) (2 x - 3)}{4 x^{2} - 6 x - 10 x + 15}$

Step 1.3.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 1.3.2.1

Group the first two terms and the last two terms.

$\frac{x - 4}{x} = \frac{(2 x + 3) (2 x - 3)}{(4 x^{2} - 6 x) - 10 x + 15}$

Step 1.3.2.2

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

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

Step 1.3.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 3$ .

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

Step 1.4

Cancel the common factor of $2 x - 3$ .

Tap for more steps...

Step 1.4.1

Cancel the common factor.

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

Step 1.4.2

Rewrite the expression.

$\frac{x - 4}{x} = \frac{2 x + 3}{2 x - 5}$

Step 2

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$(x - 4) (2 x - 5) = x (2 x + 3)$

Step 3

Solve the equation for $x$ .

Tap for more steps...

Step 3.1

Simplify $(x - 4) (2 x - 5)$ .

Tap for more steps...

Step 3.1.1

Rewrite.

$0 + 0 + (x - 4) (2 x - 5) = x (2 x + 3)$

Step 3.1.2

Simplify by adding zeros.

$(x - 4) (2 x - 5) = x (2 x + 3)$

Step 3.1.3

Expand $(x - 4) (2 x - 5)$ using the FOIL Method.

Tap for more steps...

Step 3.1.3.1

Apply the distributive property.

$x (2 x - 5) - 4 (2 x - 5) = x (2 x + 3)$

Step 3.1.3.2

Apply the distributive property.

$x (2 x) + x \cdot - 5 - 4 (2 x - 5) = x (2 x + 3)$

Step 3.1.3.3

Apply the distributive property.

$x (2 x) + x \cdot - 5 - 4 (2 x) - 4 \cdot - 5 = x (2 x + 3)$

Step 3.1.4

Simplify and combine like terms.

Tap for more steps...

Step 3.1.4.1

Simplify each term.

Tap for more steps...

Step 3.1.4.1.1

Rewrite using the commutative property of multiplication.

$2 x \cdot x + x \cdot - 5 - 4 (2 x) - 4 \cdot - 5 = x (2 x + 3)$

Step 3.1.4.1.2

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

Tap for more steps...

Step 3.1.4.1.2.1

Move $x$ .

$2 (x \cdot x) + x \cdot - 5 - 4 (2 x) - 4 \cdot - 5 = x (2 x + 3)$

Step 3.1.4.1.2.2

Multiply $x$ by $x$ .

$2 x^{2} + x \cdot - 5 - 4 (2 x) - 4 \cdot - 5 = x (2 x + 3)$

Step 3.1.4.1.3

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

$2 x^{2} - 5 \cdot x - 4 (2 x) - 4 \cdot - 5 = x (2 x + 3)$

Step 3.1.4.1.4

Multiply $2$ by $- 4$ .

$2 x^{2} - 5 x - 8 x - 4 \cdot - 5 = x (2 x + 3)$

Step 3.1.4.1.5

Multiply $- 4$ by $- 5$ .

$2 x^{2} - 5 x - 8 x + 20 = x (2 x + 3)$

Step 3.1.4.2

Subtract $8 x$ from $- 5 x$ .

$2 x^{2} - 13 x + 20 = x (2 x + 3)$

Step 3.2

Simplify $x (2 x + 3)$ .

Tap for more steps...

Step 3.2.1

Simplify by multiplying through.

Tap for more steps...

Step 3.2.1.1

Apply the distributive property.

$2 x^{2} - 13 x + 20 = x (2 x) + x \cdot 3$

Step 3.2.1.2

Reorder.

Tap for more steps...

Step 3.2.1.2.1

Rewrite using the commutative property of multiplication.

$2 x^{2} - 13 x + 20 = 2 x \cdot x + x \cdot 3$

Step 3.2.1.2.2

Move $3$ to the left of $x$ .

$2 x^{2} - 13 x + 20 = 2 x \cdot x + 3 \cdot x$

Step 3.2.2

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

Tap for more steps...

Step 3.2.2.1

Move $x$ .

$2 x^{2} - 13 x + 20 = 2 (x \cdot x) + 3 \cdot x$

Step 3.2.2.2

Multiply $x$ by $x$ .

$2 x^{2} - 13 x + 20 = 2 x^{2} + 3 \cdot x$

$2 x^{2} - 13 x + 20 = 2 x^{2} + 3 x$

Step 3.3

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

Tap for more steps...

Step 3.3.1

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

$2 x^{2} - 13 x + 20 - 2 x^{2} = 3 x$

Step 3.3.2

Subtract $3 x$ from both sides of the equation.

$2 x^{2} - 13 x + 20 - 2 x^{2} - 3 x = 0$

Step 3.3.3

Combine the opposite terms in $2 x^{2} - 13 x + 20 - 2 x^{2} - 3 x$ .

Tap for more steps...

Step 3.3.3.1

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

$- 13 x + 20 + 0 - 3 x = 0$

Step 3.3.3.2

Add $- 13 x + 20$ and $0$ .

$- 13 x + 20 - 3 x = 0$

Step 3.3.4

Subtract $3 x$ from $- 13 x$ .

$- 16 x + 20 = 0$

Step 3.4

Subtract $20$ from both sides of the equation.

$- 16 x = - 20$

Step 3.5

Divide each term in $- 16 x = - 20$ by $- 16$ and simplify.

Tap for more steps...

Step 3.5.1

Divide each term in $- 16 x = - 20$ by $- 16$ .

$\frac{- 16 x}{- 16} = \frac{- 20}{- 16}$

Step 3.5.2

Simplify the left side.

Tap for more steps...

Step 3.5.2.1

Cancel the common factor of $- 16$ .

Tap for more steps...

Step 3.5.2.1.1

Cancel the common factor.

$\frac{- 16 x}{- 16} = \frac{- 20}{- 16}$

Step 3.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 20}{- 16}$

Step 3.5.3

Simplify the right side.

Tap for more steps...

Step 3.5.3.1

Cancel the common factor of $- 20$ and $- 16$ .

Tap for more steps...

Step 3.5.3.1.1

Factor $- 4$ out of $- 20$ .

$x = \frac{- 4 (5)}{- 16}$

Step 3.5.3.1.2

Cancel the common factors.

Tap for more steps...

Step 3.5.3.1.2.1

Factor $- 4$ out of $- 16$ .

$x = \frac{- 4 \cdot 5}{- 4 \cdot 4}$

Step 3.5.3.1.2.2

Cancel the common factor.

$x = \frac{- 4 \cdot 5}{- 4 \cdot 4}$

Step 3.5.3.1.2.3

Rewrite the expression.

$x = \frac{5}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{4}$

Decimal Form:

$x = 1.25$

Mixed Number Form:

$x = 1 \frac{1}{4}$