Precalculus Examples

$y = - \frac{32}{{(20)}^{2}} x^{2} + x + 5$

Step 1

Find the x-intercepts.

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Step 1.1

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

$0 = - \frac{32}{{(20)}^{2}} x^{2} + x + 5$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- \frac{32}{20^{2}} x^{2} + x + 5 = 0$ .

$- \frac{32}{20^{2}} x^{2} + x + 5 = 0$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Raise $20$ to the power of $2$ .

$- \frac{32}{400} x^{2} + x + 5 = 0$

Step 1.2.2.2

Cancel the common factor of $32$ and $400$ .

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Step 1.2.2.2.1

Factor $16$ out of $32$ .

$- \frac{16 (2)}{400} x^{2} + x + 5 = 0$

Step 1.2.2.2.2

Cancel the common factors.

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Step 1.2.2.2.2.1

Factor $16$ out of $400$ .

$- \frac{16 \cdot 2}{16 \cdot 25} x^{2} + x + 5 = 0$

Step 1.2.2.2.2.2

Cancel the common factor.

$- \frac{16 \cdot 2}{16 \cdot 25} x^{2} + x + 5 = 0$

Step 1.2.2.2.2.3

Rewrite the expression.

$- \frac{2}{25} x^{2} + x + 5 = 0$

Step 1.2.2.3

Combine $x^{2}$ and $\frac{2}{25}$ .

$- \frac{x^{2} \cdot 2}{25} + x + 5 = 0$

Step 1.2.2.4

Move $2$ to the left of $x^{2}$ .

$- \frac{2 x^{2}}{25} + x + 5 = 0$

Step 1.2.3

Multiply through by the least common denominator $25$ , then simplify.

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Step 1.2.3.1

Apply the distributive property.

$25 (- \frac{2 x^{2}}{25}) + 25 x + 25 \cdot 5 = 0$

Step 1.2.3.2

Simplify.

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Step 1.2.3.2.1

Cancel the common factor of $25$ .

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Step 1.2.3.2.1.1

Move the leading negative in $- \frac{2 x^{2}}{25}$ into the numerator.

$25 (\frac{- 2 x^{2}}{25}) + 25 x + 25 \cdot 5 = 0$

Step 1.2.3.2.1.2

Cancel the common factor.

$25 (\frac{- 2 x^{2}}{25}) + 25 x + 25 \cdot 5 = 0$

Step 1.2.3.2.1.3

Rewrite the expression.

$- 2 x^{2} + 25 x + 25 \cdot 5 = 0$

Step 1.2.3.2.2

Multiply $25$ by $5$ .

$- 2 x^{2} + 25 x + 125 = 0$

Step 1.2.4

Use the quadratic formula to find the solutions.

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

Step 1.2.5

Substitute the values $a = - 2$ , $b = 25$ , and $c = 125$ into the quadratic formula and solve for $x$ .

$\frac{- 25 \pm \sqrt{25^{2} - 4 \cdot (- 2 \cdot 125)}}{2 \cdot - 2}$

Step 1.2.6

Simplify.

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Step 1.2.6.1

Simplify the numerator.

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Step 1.2.6.1.1

Raise $25$ to the power of $2$ .

$x = \frac{- 25 \pm \sqrt{625 - 4 \cdot - 2 \cdot 125}}{2 \cdot - 2}$

Step 1.2.6.1.2

Multiply $- 4 \cdot - 2 \cdot 125$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 25 \pm \sqrt{625 + 8 \cdot 125}}{2 \cdot - 2}$

Step 1.2.6.1.2.2

Multiply $8$ by $125$ .

$x = \frac{- 25 \pm \sqrt{625 + 1000}}{2 \cdot - 2}$

Step 1.2.6.1.3

Add $625$ and $1000$ .

$x = \frac{- 25 \pm \sqrt{1625}}{2 \cdot - 2}$

Step 1.2.6.1.4

Rewrite $1625$ as $5^{2} \cdot 65$ .

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Step 1.2.6.1.4.1

Factor $25$ out of $1625$ .

$x = \frac{- 25 \pm \sqrt{25 (65)}}{2 \cdot - 2}$

Step 1.2.6.1.4.2

Rewrite $25$ as $5^{2}$ .

$x = \frac{- 25 \pm \sqrt{5^{2} \cdot 65}}{2 \cdot - 2}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$x = \frac{- 25 \pm 5 \sqrt{65}}{2 \cdot - 2}$

Step 1.2.6.2

Multiply $2$ by $- 2$ .

$x = \frac{- 25 \pm 5 \sqrt{65}}{- 4}$

Step 1.2.6.3

Simplify $\frac{- 25 \pm 5 \sqrt{65}}{- 4}$ .

$x = \frac{25 \pm 5 \sqrt{65}}{4}$

Step 1.2.7

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

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Step 1.2.7.1

Simplify the numerator.

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Step 1.2.7.1.1

Raise $25$ to the power of $2$ .

$x = \frac{- 25 \pm \sqrt{625 - 4 \cdot - 2 \cdot 125}}{2 \cdot - 2}$

Step 1.2.7.1.2

Multiply $- 4 \cdot - 2 \cdot 125$ .

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Step 1.2.7.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 25 \pm \sqrt{625 + 8 \cdot 125}}{2 \cdot - 2}$

Step 1.2.7.1.2.2

Multiply $8$ by $125$ .

$x = \frac{- 25 \pm \sqrt{625 + 1000}}{2 \cdot - 2}$

Step 1.2.7.1.3

Add $625$ and $1000$ .

$x = \frac{- 25 \pm \sqrt{1625}}{2 \cdot - 2}$

Step 1.2.7.1.4

Rewrite $1625$ as $5^{2} \cdot 65$ .

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Step 1.2.7.1.4.1

Factor $25$ out of $1625$ .

$x = \frac{- 25 \pm \sqrt{25 (65)}}{2 \cdot - 2}$

Step 1.2.7.1.4.2

Rewrite $25$ as $5^{2}$ .

$x = \frac{- 25 \pm \sqrt{5^{2} \cdot 65}}{2 \cdot - 2}$

Step 1.2.7.1.5

Pull terms out from under the radical.

$x = \frac{- 25 \pm 5 \sqrt{65}}{2 \cdot - 2}$

Step 1.2.7.2

Multiply $2$ by $- 2$ .

$x = \frac{- 25 \pm 5 \sqrt{65}}{- 4}$

Step 1.2.7.3

Simplify $\frac{- 25 \pm 5 \sqrt{65}}{- 4}$ .

$x = \frac{25 \pm 5 \sqrt{65}}{4}$

Step 1.2.7.4

Change the $\pm$ to $+$ .

$x = \frac{25 + 5 \sqrt{65}}{4}$

Step 1.2.8

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

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Step 1.2.8.1

Simplify the numerator.

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Step 1.2.8.1.1

Raise $25$ to the power of $2$ .

$x = \frac{- 25 \pm \sqrt{625 - 4 \cdot - 2 \cdot 125}}{2 \cdot - 2}$

Step 1.2.8.1.2

Multiply $- 4 \cdot - 2 \cdot 125$ .

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Step 1.2.8.1.2.1

Multiply $- 4$ by $- 2$ .

$x = \frac{- 25 \pm \sqrt{625 + 8 \cdot 125}}{2 \cdot - 2}$

Step 1.2.8.1.2.2

Multiply $8$ by $125$ .

$x = \frac{- 25 \pm \sqrt{625 + 1000}}{2 \cdot - 2}$

Step 1.2.8.1.3

Add $625$ and $1000$ .

$x = \frac{- 25 \pm \sqrt{1625}}{2 \cdot - 2}$

Step 1.2.8.1.4

Rewrite $1625$ as $5^{2} \cdot 65$ .

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Step 1.2.8.1.4.1

Factor $25$ out of $1625$ .

$x = \frac{- 25 \pm \sqrt{25 (65)}}{2 \cdot - 2}$

Step 1.2.8.1.4.2

Rewrite $25$ as $5^{2}$ .

$x = \frac{- 25 \pm \sqrt{5^{2} \cdot 65}}{2 \cdot - 2}$

Step 1.2.8.1.5

Pull terms out from under the radical.

$x = \frac{- 25 \pm 5 \sqrt{65}}{2 \cdot - 2}$

Step 1.2.8.2

Multiply $2$ by $- 2$ .

$x = \frac{- 25 \pm 5 \sqrt{65}}{- 4}$

Step 1.2.8.3

Simplify $\frac{- 25 \pm 5 \sqrt{65}}{- 4}$ .

$x = \frac{25 \pm 5 \sqrt{65}}{4}$

Step 1.2.8.4

Change the $\pm$ to $-$ .

$x = \frac{25 - 5 \sqrt{65}}{4}$

Step 1.2.9

The final answer is the combination of both solutions.

$x = \frac{25 + 5 \sqrt{65}}{4}, \frac{25 - 5 \sqrt{65}}{4}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\frac{25 + 5 \sqrt{65}}{4}, 0), (\frac{25 - 5 \sqrt{65}}{4}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

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

$y = - \frac{32}{{(20)}^{2}} \cdot {(0)}^{2} + 0 + 5$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - \frac{32}{{(20)}^{2}} \cdot {(0)}^{2} + 0 + 5$

Step 2.2.2

Remove parentheses.

$y = - \frac{32}{20^{2}} \cdot {(0)}^{2} + 0 + 5$

Step 2.2.3

Remove parentheses.

$y = - \frac{32}{20^{2}} \cdot 0^{2} + 0 + 5$

Step 2.2.4

Remove parentheses.

$y = - \frac{32}{{(20)}^{2}} \cdot {(0)}^{2} + 0 + 5$

Step 2.2.5

Simplify $- \frac{32}{{(20)}^{2}} \cdot {(0)}^{2} + 0 + 5$ .

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Step 2.2.5.1

Simplify each term.

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Step 2.2.5.1.1

Raise $20$ to the power of $2$ .

$y = - \frac{32}{400} \cdot {(0)}^{2} + 0 + 5$

Step 2.2.5.1.2

Cancel the common factor of $32$ and $400$ .

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Step 2.2.5.1.2.1

Factor $16$ out of $32$ .

$y = - \frac{16 (2)}{400} \cdot {(0)}^{2} + 0 + 5$

Step 2.2.5.1.2.2

Cancel the common factors.

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Step 2.2.5.1.2.2.1

Factor $16$ out of $400$ .

$y = - \frac{16 \cdot 2}{16 \cdot 25} \cdot {(0)}^{2} + 0 + 5$

Step 2.2.5.1.2.2.2

Cancel the common factor.

$y = - \frac{16 \cdot 2}{16 \cdot 25} \cdot {(0)}^{2} + 0 + 5$

Step 2.2.5.1.2.2.3

Rewrite the expression.

$y = - \frac{2}{25} \cdot {(0)}^{2} + 0 + 5$

Step 2.2.5.1.3

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

$y = - \frac{2}{25} \cdot 0 + 0 + 5$

Step 2.2.5.1.4

Multiply $- \frac{2}{25} \cdot 0$ .

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Step 2.2.5.1.4.1

Multiply $0$ by $- 1$ .

$y = 0 (\frac{2}{25}) + 0 + 5$

Step 2.2.5.1.4.2

Multiply $0$ by $\frac{2}{25}$ .

$y = 0 + 0 + 5$

Step 2.2.5.2

Simplify by adding numbers.

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Step 2.2.5.2.1

Add $0$ and $0$ .

$y = 0 + 5$

Step 2.2.5.2.2

Add $0$ and $5$ .

$y = 5$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 5)$

Step 3

List the intersections.

x-intercept(s): $(\frac{25 + 5 \sqrt{65}}{4}, 0), (\frac{25 - 5 \sqrt{65}}{4}, 0)$

y-intercept(s): $(0, 5)$

Step 4