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

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

Step 1

Find the x-intercepts.

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

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

Solve the equation.

Set the numerator equal to zero.

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

Solve the equation for $x$ .

Subtract $5$ from both sides of the equation.

$2 x^{2} = - 5$

Divide each term in $2 x^{2} = - 5$ by $2$ and simplify.

Divide each term in $2 x^{2} = - 5$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x^{2}$ by $1$ .

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

Simplify the right side.

Move the negative in front of the fraction.

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

Take the square root of both sides of the equation to eliminate the exponent on the left side.

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

Simplify $\pm \sqrt{- \frac{5}{2}}$ .

Rewrite $- 1$ as $i^{2}$ .

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

Pull terms out from under the radical.

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

Rewrite $\sqrt{\frac{5}{2}}$ as $\frac{\sqrt{5}}{\sqrt{2}}$ .

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

Multiply $\frac{\sqrt{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Combine and simplify the denominator.

Multiply $\frac{\sqrt{5}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Raise $\sqrt{2}$ to the power of $1$ .

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

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

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

$x = \pm i \frac{\sqrt{5} \sqrt{2}}{2^{1}}$

Evaluate the exponent.

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

Simplify the numerator.

Combine using the product rule for radicals.

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

Multiply $5$ by $2$ .

$x = \pm i \frac{\sqrt{10}}{2}$

Combine $i$ and $\frac{\sqrt{10}}{2}$ .

$x = \pm \frac{i \sqrt{10}}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{i \sqrt{10}}{2}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{i \sqrt{10}}{2}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{i \sqrt{10}}{2}, - \frac{i \sqrt{10}}{2}$

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.

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

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

The equation has an undefined fraction.

Undefined

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

y-intercept(s): $None$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $None$

Step 4

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

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