Precalculus Examples

$f (x) = \frac{2 x^{2} - 5 x + 5}{x - 2}$

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 $x$ .

$0 = \frac{2 x^{2} - 5 x + 5}{x - 2}$

Step 1.2

Solve the equation.

Tap for more steps...

Step 1.2.1

Set the numerator equal to zero.

$2 x^{2} - 5 x + 5 = 0$

Step 1.2.2

Solve the equation for $x$ .

Tap for more steps...

Step 1.2.2.1

Use the quadratic formula to find the solutions.

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

Step 1.2.2.2

Substitute the values $a = 2$ , $b = - 5$ , and $c = 5$ into the quadratic formula and solve for $x$ .

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

Step 1.2.2.3

Simplify.

Tap for more steps...

Step 1.2.2.3.1

Simplify the numerator.

Tap for more steps...

Step 1.2.2.3.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 2 \cdot 5}}{2 \cdot 2}$

Step 1.2.2.3.1.2

Multiply $- 4 \cdot 2 \cdot 5$ .

Tap for more steps...

Step 1.2.2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 8 \cdot 5}}{2 \cdot 2}$

Step 1.2.2.3.1.2.2

Multiply $- 8$ by $5$ .

$x = \frac{5 \pm \sqrt{25 - 40}}{2 \cdot 2}$

Step 1.2.2.3.1.3

Subtract $40$ from $25$ .

$x = \frac{5 \pm \sqrt{- 15}}{2 \cdot 2}$

Step 1.2.2.3.1.4

Rewrite $- 15$ as $- 1 (15)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 2}$

Step 1.2.2.3.1.5

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 2}$

Step 1.2.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \sqrt{15}}{2 \cdot 2}$

Step 1.2.2.3.2

Multiply $2$ by $2$ .

$x = \frac{5 \pm i \sqrt{15}}{4}$

Step 1.2.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Tap for more steps...

Step 1.2.2.4.1

Simplify the numerator.

Tap for more steps...

Step 1.2.2.4.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 2 \cdot 5}}{2 \cdot 2}$

Step 1.2.2.4.1.2

Multiply $- 4 \cdot 2 \cdot 5$ .

Tap for more steps...

Step 1.2.2.4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 8 \cdot 5}}{2 \cdot 2}$

Step 1.2.2.4.1.2.2

Multiply $- 8$ by $5$ .

$x = \frac{5 \pm \sqrt{25 - 40}}{2 \cdot 2}$

Step 1.2.2.4.1.3

Subtract $40$ from $25$ .

$x = \frac{5 \pm \sqrt{- 15}}{2 \cdot 2}$

Step 1.2.2.4.1.4

Rewrite $- 15$ as $- 1 (15)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 2}$

Step 1.2.2.4.1.5

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 2}$

Step 1.2.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \sqrt{15}}{2 \cdot 2}$

Step 1.2.2.4.2

Multiply $2$ by $2$ .

$x = \frac{5 \pm i \sqrt{15}}{4}$

Step 1.2.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{5 + i \sqrt{15}}{4}$

Step 1.2.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Tap for more steps...

Step 1.2.2.5.1

Simplify the numerator.

Tap for more steps...

Step 1.2.2.5.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 2 \cdot 5}}{2 \cdot 2}$

Step 1.2.2.5.1.2

Multiply $- 4 \cdot 2 \cdot 5$ .

Tap for more steps...

Step 1.2.2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 8 \cdot 5}}{2 \cdot 2}$

Step 1.2.2.5.1.2.2

Multiply $- 8$ by $5$ .

$x = \frac{5 \pm \sqrt{25 - 40}}{2 \cdot 2}$

Step 1.2.2.5.1.3

Subtract $40$ from $25$ .

$x = \frac{5 \pm \sqrt{- 15}}{2 \cdot 2}$

Step 1.2.2.5.1.4

Rewrite $- 15$ as $- 1 (15)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 15}}{2 \cdot 2}$

Step 1.2.2.5.1.5

Rewrite $\sqrt{- 1 (15)}$ as $\sqrt{- 1} \cdot \sqrt{15}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{15}}{2 \cdot 2}$

Step 1.2.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{5 \pm i \sqrt{15}}{2 \cdot 2}$

Step 1.2.2.5.2

Multiply $2$ by $2$ .

$x = \frac{5 \pm i \sqrt{15}}{4}$

Step 1.2.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{5 - i \sqrt{15}}{4}$

Step 1.2.2.6

The final answer is the combination of both solutions.

$x = \frac{5 + i \sqrt{15}}{4}, \frac{5 - i \sqrt{15}}{4}$

Step 1.3

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

x-intercept(s): $None$

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 $y$ .

$y = \frac{2 {(0)}^{2} - 5 \cdot 0 + 5}{(0) - 2}$

Step 2.2

Solve the equation.

Tap for more steps...

Step 2.2.1

Remove parentheses.

$y = \frac{2 \cdot 0^{2} - 5 \cdot 0 + 5}{(0) - 2}$

Step 2.2.2

Remove parentheses.

$y = \frac{2 \cdot 0^{2} - 5 \cdot 0 + 5}{0 - 2}$

Step 2.2.3

Remove parentheses.

$y = \frac{2 {(0)}^{2} - 5 \cdot 0 + 5}{(0) - 2}$

Step 2.2.4

Simplify $\frac{2 {(0)}^{2} - 5 \cdot 0 + 5}{(0) - 2}$ .

Tap for more steps...

Step 2.2.4.1

Simplify the numerator.

Tap for more steps...

Step 2.2.4.1.1

Raising $0$ to any positive power yields $0$ .

$y = \frac{2 \cdot 0 - 5 \cdot 0 + 5}{0 - 2}$

Step 2.2.4.1.2

Multiply $2$ by $0$ .

$y = \frac{0 - 5 \cdot 0 + 5}{0 - 2}$

Step 2.2.4.1.3

Multiply $- 5$ by $0$ .

$y = \frac{0 + 0 + 5}{0 - 2}$

Step 2.2.4.1.4

Add $0$ and $0$ .

$y = \frac{0 + 5}{0 - 2}$

Step 2.2.4.1.5

Add $0$ and $5$ .

$y = \frac{5}{0 - 2}$

Step 2.2.4.2

Simplify the expression.

Tap for more steps...

Step 2.2.4.2.1

Subtract $2$ from $0$ .

$y = \frac{5}{- 2}$

Step 2.2.4.2.2

Move the negative in front of the fraction.

$y = - \frac{5}{2}$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, - \frac{5}{2})$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - \frac{5}{2})$

Step 4