Precalculus Examples

$(- \frac{5 - \sqrt{19}}{6}, 0)$ , $(- \frac{5 + \sqrt{19}}{6}, 0)$

Step 1

Find the slope of the line between $(- \frac{5 - \sqrt{19}}{6}, 0)$ and $(- \frac{5 + \sqrt{19}}{6}, 0)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

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Step 1.1

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

Step 1.2

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Step 1.3

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{0 - (0)}{- \frac{5 + \sqrt{19}}{6} - (- \frac{5 - \sqrt{19}}{6})}$

Step 1.4

Simplify.

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Step 1.4.1

Multiply the numerator and denominator of the fraction by $6$ .

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Step 1.4.1.1

Multiply $\frac{0 - (0)}{- \frac{5 + \sqrt{19}}{6} - - \frac{5 - \sqrt{19}}{6}}$ by $\frac{6}{6}$ .

$m = \frac{6}{6} \cdot \frac{0 - (0)}{- \frac{5 + \sqrt{19}}{6} + \frac{5 - \sqrt{19}}{6}}$

Step 1.4.1.2

Combine.

$m = \frac{6 (0 - (0))}{6 (- \frac{5 + \sqrt{19}}{6} + \frac{5 - \sqrt{19}}{6})}$

Step 1.4.2

Apply the distributive property.

$m = \frac{6 \cdot 0 + 6 (- (0))}{6 (- \frac{5 + \sqrt{19}}{6}) + 6 (\frac{5 - \sqrt{19}}{6})}$

Step 1.4.3

Cancel the common factor of $6$ .

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Step 1.4.3.1

Move the leading negative in $- \frac{5 + \sqrt{19}}{6}$ into the numerator.

$m = \frac{6 \cdot 0 + 6 (- (0))}{6 (\frac{- (5 + \sqrt{19})}{6}) + 6 (\frac{5 - \sqrt{19}}{6})}$

Step 1.4.3.2

Cancel the common factor.

$m = \frac{6 \cdot 0 + 6 (- (0))}{6 (\frac{- (5 + \sqrt{19})}{6}) + 6 (\frac{5 - \sqrt{19}}{6})}$

Step 1.4.3.3

Rewrite the expression.

$m = \frac{6 \cdot 0 + 6 (- (0))}{- (5 + \sqrt{19}) + 6 (\frac{5 - \sqrt{19}}{6})}$

Step 1.4.4

Simplify the numerator.

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Step 1.4.4.1

Multiply $6$ by $0$ .

$m = \frac{0 + 6 (- (0))}{- (5 + \sqrt{19}) + 6 (\frac{5 - \sqrt{19}}{6})}$

Step 1.4.4.2

Multiply $6 (- (0))$ .

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Step 1.4.4.2.1

Multiply $- 1$ by $0$ .

$m = \frac{0 + 6 \cdot 0}{- (5 + \sqrt{19}) + 6 (\frac{5 - \sqrt{19}}{6})}$

Step 1.4.4.2.2

Multiply $6$ by $0$ .

$m = \frac{0 + 0}{- (5 + \sqrt{19}) + 6 (\frac{5 - \sqrt{19}}{6})}$

Step 1.4.4.3

Add $0$ and $0$ .

$m = \frac{0}{- (5 + \sqrt{19}) + 6 (\frac{5 - \sqrt{19}}{6})}$

Step 1.4.5

Simplify the denominator.

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Step 1.4.5.1

Apply the distributive property.

$m = \frac{0}{- 1 \cdot 5 - \sqrt{19} + 6 (\frac{5 - \sqrt{19}}{6})}$

Step 1.4.5.2

Multiply $- 1$ by $5$ .

$m = \frac{0}{- 5 - \sqrt{19} + 6 (\frac{5 - \sqrt{19}}{6})}$

Step 1.4.5.3

Multiply $- - \frac{5 - \sqrt{19}}{6}$ .

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Step 1.4.5.3.1

Multiply $- 1$ by $- 1$ .

$m = \frac{0}{- 5 - \sqrt{19} + 6 (1 (\frac{5 - \sqrt{19}}{6}))}$

Step 1.4.5.3.2

Multiply $\frac{5 - \sqrt{19}}{6}$ by $1$ .

$m = \frac{0}{- 5 - \sqrt{19} + 6 (\frac{5 - \sqrt{19}}{6})}$

Step 1.4.5.4

Cancel the common factor of $6$ .

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Step 1.4.5.4.1

Cancel the common factor.

$m = \frac{0}{- 5 - \sqrt{19} + 6 (\frac{5 - \sqrt{19}}{6})}$

Step 1.4.5.4.2

Rewrite the expression.

$m = \frac{0}{- 5 - \sqrt{19} + 5 - \sqrt{19}}$

Step 1.4.5.5

Add $- 5$ and $5$ .

$m = \frac{0}{0 - \sqrt{19} - \sqrt{19}}$

Step 1.4.5.6

Subtract $\sqrt{19}$ from $0$ .

$m = \frac{0}{- \sqrt{19} - \sqrt{19}}$

Step 1.4.5.7

Subtract $\sqrt{19}$ from $- \sqrt{19}$ .

$m = \frac{0}{- 2 \sqrt{19}}$

Step 1.4.6

Reduce the expression by cancelling the common factors.

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Step 1.4.6.1

Cancel the common factor of $0$ and $- 2$ .

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Step 1.4.6.1.1

Factor $2$ out of $0$ .

$m = \frac{2 (0)}{- 2 \sqrt{19}}$

Step 1.4.6.1.2

Cancel the common factors.

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Step 1.4.6.1.2.1

Factor $2$ out of $- 2 \sqrt{19}$ .

$m = \frac{2 (0)}{2 (- \sqrt{19})}$

Step 1.4.6.1.2.2

Cancel the common factor.

$m = \frac{2 \cdot 0}{2 (- \sqrt{19})}$

Step 1.4.6.1.2.3

Rewrite the expression.

$m = \frac{0}{- \sqrt{19}}$

Step 1.4.6.2

Divide $0$ by $- \sqrt{19}$ .

$m = 0$

Step 2

Use the slope $0$ and a given point $(- \frac{5 - \sqrt{19}}{6}, 0)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (0) = 0 \cdot (x - (- \frac{5 - \sqrt{19}}{6}))$

Step 3

Simplify the equation and keep it in point-slope form.

$y + 0 = 0 \cdot (x + \frac{5 - \sqrt{19}}{6})$

Step 4

Solve for $y$ .

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Step 4.1

Add $y$ and $0$ .

$y = 0 \cdot (x + \frac{5 - \sqrt{19}}{6})$

Step 4.2

Simplify $0 \cdot (x + \frac{5 - \sqrt{19}}{6})$ .

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Step 4.2.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$y = 0 \cdot (x \cdot \frac{6}{6} + \frac{5 - \sqrt{19}}{6})$

Step 4.2.2

Simplify terms.

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Step 4.2.2.1

Combine $x$ and $\frac{6}{6}$ .

$y = 0 \cdot (\frac{x \cdot 6}{6} + \frac{5 - \sqrt{19}}{6})$

Step 4.2.2.2

Combine the numerators over the common denominator.

$y = 0 \cdot \frac{x \cdot 6 + 5 - \sqrt{19}}{6}$

Step 4.2.3

Move $6$ to the left of $x$ .

$y = 0 \cdot \frac{6 x + 5 - \sqrt{19}}{6}$

Step 4.2.4

Multiply $0$ by $\frac{6 x + 5 - \sqrt{19}}{6}$ .

$y = 0$

Step 5

List the equation in different forms.

Slope-intercept form:

$y = 0$

Point-slope form:

$y + 0 = 0 \cdot (x + \frac{5 - \sqrt{19}}{6})$

Step 6