Precalculus Examples

$\frac{x^{2} - x + 1}{x - 1}$

Step 1

Write $\frac{x^{2} - x + 1}{x - 1}$ as an equation.

$y = \frac{x^{2} - x + 1}{x - 1}$

Step 2

Find the x-intercepts.

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Step 2.1

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

$0 = \frac{x^{2} - x + 1}{x - 1}$

Step 2.2

Solve the equation.

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Step 2.2.1

Set the numerator equal to zero.

$x^{2} - x + 1 = 0$

Step 2.2.2

Solve the equation for $x$ .

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Step 2.2.2.1

Use the quadratic formula to find the solutions.

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

Step 2.2.2.2

Substitute the values $a = 1$ , $b = - 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 2.2.2.3

Simplify.

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Step 2.2.2.3.1

Simplify the numerator.

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Step 2.2.2.3.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.2.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 2.2.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.2.2.3.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 2.2.2.3.1.3

Subtract $4$ from $1$ .

$x = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 2.2.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 2.2.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 2.2.2.3.1.6

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 2.2.2.3.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

Step 2.2.2.4

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

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Step 2.2.2.4.1

Simplify the numerator.

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Step 2.2.2.4.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.2.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 2.2.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.2.2.4.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 2.2.2.4.1.3

Subtract $4$ from $1$ .

$x = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 2.2.2.4.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 2.2.2.4.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 2.2.2.4.1.6

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 2.2.2.4.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

Step 2.2.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{1 + i \sqrt{3}}{2}$

Step 2.2.2.5

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

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Step 2.2.2.5.1

Simplify the numerator.

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Step 2.2.2.5.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 2.2.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 2.2.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 2.2.2.5.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 2.2.2.5.1.3

Subtract $4$ from $1$ .

$x = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 2.2.2.5.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 2.2.2.5.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 2.2.2.5.1.6

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 2.2.2.5.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

Step 2.2.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{1 - i \sqrt{3}}{2}$

Step 2.2.2.6

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

Step 2.3

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.

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Step 3.1

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

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

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

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

Step 3.2.2

Remove parentheses.

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

Step 3.2.3

Remove parentheses.

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

Step 3.2.4

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

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Step 3.2.4.1

Simplify the numerator.

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Step 3.2.4.1.1

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

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

Step 3.2.4.1.2

Multiply $- 1$ by $0$ .

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

Step 3.2.4.1.3

Add $0$ and $0$ .

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

Step 3.2.4.1.4

Add $0$ and $1$ .

$y = \frac{1}{0 - 1}$

Step 3.2.4.2

Simplify the expression.

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Step 3.2.4.2.1

Subtract $1$ from $0$ .

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

Step 3.2.4.2.2

Divide $1$ by $- 1$ .

$y = - 1$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 1)$

Step 4

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - 1)$

Step 5