Algebra Examples

$f (x) = \frac{(4 x - 3) (x - 1)}{(2 x + 1) (x + 1)}$

Step 1

Find the x-intercepts.

Tap for more steps...

Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \frac{(4 x - 3) (x - 1)}{(2 x + 1) (x + 1)}$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Set the numerator equal to zero.

$(4 x - 3) (x - 1) = 0$

Step 1.2.2

Solve the equation for $x$ .

Tap for more steps...

Step 1.2.2.1

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

$4 x - 3 = 0$

$x - 1 = 0$

Step 1.2.2.2

Set $4 x - 3$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.2.2.1

Set $4 x - 3$ equal to $0$ .

$4 x - 3 = 0$

Step 1.2.2.2.2

Solve $4 x - 3 = 0$ for $x$ .

Tap for more steps...

Step 1.2.2.2.2.1

Add $3$ to both sides of the equation.

$4 x = 3$

Step 1.2.2.2.2.2

Divide each term in $4 x = 3$ by $4$ and simplify.

Tap for more steps...

Step 1.2.2.2.2.2.1

Divide each term in $4 x = 3$ by $4$ .

$\frac{4 x}{4} = \frac{3}{4}$

Step 1.2.2.2.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.2.2.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.2.2.2.2.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{3}{4}$

Step 1.2.2.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{3}{4}$

Step 1.2.2.3

Set $x - 1$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.2.3.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 1.2.2.3.2

Add $1$ to both sides of the equation.

$x = 1$

Step 1.2.2.4

The final solution is all the values that make $(4 x - 3) (x - 1) = 0$ true.

$x = \frac{3}{4}, 1$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{3}{4}, 0), (1, 0)$

Step 2

Find the y-intercepts.

Tap for more steps...

Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \frac{(4 (0) - 3) ((0) - 1)}{(2 (0) + 1) ((0) + 1)}$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = \frac{(4 (0) - 3) (0 - 1)}{(2 (0) + 1) ((0) + 1)}$

Step 2.2.2

Remove parentheses.

$y = \frac{(4 (0) - 3) (0 - 1)}{(2 (0) + 1) (0 + 1)}$

Step 2.2.3

Remove parentheses.

$y = \frac{(4 (0) - 3) ((0) - 1)}{(2 (0) + 1) ((0) + 1)}$

Step 2.2.4

Simplify $\frac{(4 (0) - 3) ((0) - 1)}{(2 (0) + 1) ((0) + 1)}$ .

Tap for more steps...

Step 2.2.4.1

Cancel the common factor of $0 - 1$ and $0 + 1$ .

Tap for more steps...

Step 2.2.4.1.1

Rewrite $0$ as $- 1 (0)$ .

$y = \frac{(4 (0) - 3) (- 1 (0) - 1)}{(2 (0) + 1) (0 + 1)}$

Step 2.2.4.1.2

Rewrite $- 1$ as $- 1 (1)$ .

$y = \frac{(4 (0) - 3) (- 1 \cdot 0 - 1 \cdot 1)}{(2 (0) + 1) (0 + 1)}$

Step 2.2.4.1.3

Factor $- 1$ out of $- 1 \cdot 0 - 1 \cdot 1$ .

$y = \frac{(4 (0) - 3) (- 1 \cdot (0 + 1))}{(2 (0) + 1) (0 + 1)}$

Step 2.2.4.1.4

Cancel the common factor.

$y = \frac{(4 (0) - 3) (- 1 \cdot (0 + 1))}{(2 (0) + 1) (0 + 1)}$

Step 2.2.4.1.5

Rewrite the expression.

$y = \frac{(4 (0) - 3) \cdot (- 1)}{2 (0) + 1}$

Step 2.2.4.2

Simplify the numerator.

Tap for more steps...

Step 2.2.4.2.1

Multiply $4$ by $0$ .

$y = \frac{(0 - 3) \cdot - 1}{2 (0) + 1}$

Step 2.2.4.2.2

Subtract $3$ from $0$ .

$y = \frac{- 3 \cdot - 1}{2 (0) + 1}$

Step 2.2.4.3

Simplify the denominator.

Tap for more steps...

Step 2.2.4.3.1

Multiply $2$ by $0$ .

$y = \frac{- 3 \cdot - 1}{0 + 1}$

Step 2.2.4.3.2

Add $0$ and $1$ .

$y = \frac{- 3 \cdot - 1}{1}$

Step 2.2.4.4

Simplify the expression.

Tap for more steps...

Step 2.2.4.4.1

Multiply $- 3$ by $- 1$ .

$y = \frac{3}{1}$

Step 2.2.4.4.2

Divide $3$ by $1$ .

$y = 3$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 3)$

Step 3

List the intersections.

x-intercept(s): $(\frac{3}{4}, 0), (1, 0)$

y-intercept(s): $(0, 3)$

Step 4