Find the x and y Intercepts (2x^2-5x+5)/(x-2)

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

Step 1

Write $\frac{2 x^{2} - 5 x + 5}{x - 2}$ as an equation.

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

Step 2

Find the x-intercepts.

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

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

Solve the equation.

Set the numerator equal to zero.

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

Solve the equation for $x$ .

Use the quadratic formula to find the solutions.

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

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

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

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $2$ .

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

Multiply $- 8$ by $5$ .

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

Subtract $40$ from $25$ .

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

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

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

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

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

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

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

Multiply $2$ by $2$ .

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

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

Simplify the numerator.

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

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

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

Multiply $- 4$ by $2$ .

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

Multiply $- 8$ by $5$ .

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

Subtract $40$ from $25$ .

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

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

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

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

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

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

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

Multiply $2$ by $2$ .

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

Change the $\pm$ to $+$ .

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

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

Simplify the numerator.

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

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

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

Multiply $- 4$ by $2$ .

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

Multiply $- 8$ by $5$ .

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

Subtract $40$ from $25$ .

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

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

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

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

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

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

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

Multiply $2$ by $2$ .

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

Change the $\pm$ to $-$ .

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

The final answer is the combination of both solutions.

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

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

x-intercept(s): $None$

Step 3

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

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

Remove parentheses.

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

Remove parentheses.

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

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

Simplify the numerator.

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

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

Multiply $2$ by $0$ .

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

Multiply $- 5$ by $0$ .

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

Add $0$ and $0$ .

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

Add $0$ and $5$ .

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

Simplify the expression.

Subtract $2$ from $0$ .

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

Move the negative in front of the fraction.

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

y-intercept(s) in point form.

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

Step 4

List the intersections.

x-intercept(s): $None$

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

Step 5

Find the x and y Intercepts (2x^2-5x+5)/(x-2)

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