Algebra Examples

$h (x) = - \frac{1}{64} x^{2} + \frac{13}{32} x + 2$

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 $x$ .

$0 = - \frac{1}{64} x^{2} + \frac{13}{32} x + 2$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- \frac{1}{64} x^{2} + \frac{13}{32} x + 2 = 0$ .

$- \frac{1}{64} x^{2} + \frac{13}{32} x + 2 = 0$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Combine $x^{2}$ and $\frac{1}{64}$ .

$- \frac{x^{2}}{64} + \frac{13}{32} x + 2 = 0$

Step 1.2.2.2

Combine $\frac{13}{32}$ and $x$ .

$- \frac{x^{2}}{64} + \frac{13 x}{32} + 2 = 0$

Step 1.2.3

Multiply through by the least common denominator $64$ , then simplify.

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Step 1.2.3.1

Apply the distributive property.

$64 (- \frac{x^{2}}{64}) + 64 (\frac{13 x}{32}) + 64 \cdot 2 = 0$

Step 1.2.3.2

Simplify.

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Step 1.2.3.2.1

Cancel the common factor of $64$ .

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Step 1.2.3.2.1.1

Move the leading negative in $- \frac{x^{2}}{64}$ into the numerator.

$64 (\frac{- x^{2}}{64}) + 64 (\frac{13 x}{32}) + 64 \cdot 2 = 0$

Step 1.2.3.2.1.2

Cancel the common factor.

$64 (\frac{- x^{2}}{64}) + 64 (\frac{13 x}{32}) + 64 \cdot 2 = 0$

Step 1.2.3.2.1.3

Rewrite the expression.

$- x^{2} + 64 (\frac{13 x}{32}) + 64 \cdot 2 = 0$

Step 1.2.3.2.2

Cancel the common factor of $32$ .

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Step 1.2.3.2.2.1

Factor $32$ out of $64$ .

$- x^{2} + 32 (2) (\frac{13 x}{32}) + 64 \cdot 2 = 0$

Step 1.2.3.2.2.2

Cancel the common factor.

$- x^{2} + 32 \cdot (2 (\frac{13 x}{32})) + 64 \cdot 2 = 0$

Step 1.2.3.2.2.3

Rewrite the expression.

$- x^{2} + 2 (13 x) + 64 \cdot 2 = 0$

Step 1.2.3.2.3

Multiply $13$ by $2$ .

$- x^{2} + 26 x + 64 \cdot 2 = 0$

Step 1.2.3.2.4

Multiply $64$ by $2$ .

$- x^{2} + 26 x + 128 = 0$

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 $a = - 1$ , $b = 26$ , and $c = 128$ into the quadratic formula and solve for $x$ .

$\frac{- 26 \pm \sqrt{26^{2} - 4 \cdot (- 1 \cdot 128)}}{2 \cdot - 1}$

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 $26$ to the power of $2$ .

$x = \frac{- 26 \pm \sqrt{676 - 4 \cdot - 1 \cdot 128}}{2 \cdot - 1}$

Step 1.2.6.1.2

Multiply $- 4 \cdot - 1 \cdot 128$ .

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Step 1.2.6.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 26 \pm \sqrt{676 + 4 \cdot 128}}{2 \cdot - 1}$

Step 1.2.6.1.2.2

Multiply $4$ by $128$ .

$x = \frac{- 26 \pm \sqrt{676 + 512}}{2 \cdot - 1}$

Step 1.2.6.1.3

Add $676$ and $512$ .

$x = \frac{- 26 \pm \sqrt{1188}}{2 \cdot - 1}$

Step 1.2.6.1.4

Rewrite $1188$ as $6^{2} \cdot 33$ .

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Step 1.2.6.1.4.1

Factor $36$ out of $1188$ .

$x = \frac{- 26 \pm \sqrt{36 (33)}}{2 \cdot - 1}$

Step 1.2.6.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 26 \pm \sqrt{6^{2} \cdot 33}}{2 \cdot - 1}$

Step 1.2.6.1.5

Pull terms out from under the radical.

$x = \frac{- 26 \pm 6 \sqrt{33}}{2 \cdot - 1}$

Step 1.2.6.2

Multiply $2$ by $- 1$ .

$x = \frac{- 26 \pm 6 \sqrt{33}}{- 2}$

Step 1.2.6.3

Simplify $\frac{- 26 \pm 6 \sqrt{33}}{- 2}$ .

$x = 13 \pm 3 \sqrt{33}$

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 $26$ to the power of $2$ .

$x = \frac{- 26 \pm \sqrt{676 - 4 \cdot - 1 \cdot 128}}{2 \cdot - 1}$

Step 1.2.7.1.2

Multiply $- 4 \cdot - 1 \cdot 128$ .

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Step 1.2.7.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 26 \pm \sqrt{676 + 4 \cdot 128}}{2 \cdot - 1}$

Step 1.2.7.1.2.2

Multiply $4$ by $128$ .

$x = \frac{- 26 \pm \sqrt{676 + 512}}{2 \cdot - 1}$

Step 1.2.7.1.3

Add $676$ and $512$ .

$x = \frac{- 26 \pm \sqrt{1188}}{2 \cdot - 1}$

Step 1.2.7.1.4

Rewrite $1188$ as $6^{2} \cdot 33$ .

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Step 1.2.7.1.4.1

Factor $36$ out of $1188$ .

$x = \frac{- 26 \pm \sqrt{36 (33)}}{2 \cdot - 1}$

Step 1.2.7.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 26 \pm \sqrt{6^{2} \cdot 33}}{2 \cdot - 1}$

Step 1.2.7.1.5

Pull terms out from under the radical.

$x = \frac{- 26 \pm 6 \sqrt{33}}{2 \cdot - 1}$

Step 1.2.7.2

Multiply $2$ by $- 1$ .

$x = \frac{- 26 \pm 6 \sqrt{33}}{- 2}$

Step 1.2.7.3

Simplify $\frac{- 26 \pm 6 \sqrt{33}}{- 2}$ .

$x = 13 \pm 3 \sqrt{33}$

Step 1.2.7.4

Change the $\pm$ to $+$ .

$x = 13 + 3 \sqrt{33}$

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 $26$ to the power of $2$ .

$x = \frac{- 26 \pm \sqrt{676 - 4 \cdot - 1 \cdot 128}}{2 \cdot - 1}$

Step 1.2.8.1.2

Multiply $- 4 \cdot - 1 \cdot 128$ .

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Step 1.2.8.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{- 26 \pm \sqrt{676 + 4 \cdot 128}}{2 \cdot - 1}$

Step 1.2.8.1.2.2

Multiply $4$ by $128$ .

$x = \frac{- 26 \pm \sqrt{676 + 512}}{2 \cdot - 1}$

Step 1.2.8.1.3

Add $676$ and $512$ .

$x = \frac{- 26 \pm \sqrt{1188}}{2 \cdot - 1}$

Step 1.2.8.1.4

Rewrite $1188$ as $6^{2} \cdot 33$ .

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Step 1.2.8.1.4.1

Factor $36$ out of $1188$ .

$x = \frac{- 26 \pm \sqrt{36 (33)}}{2 \cdot - 1}$

Step 1.2.8.1.4.2

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 26 \pm \sqrt{6^{2} \cdot 33}}{2 \cdot - 1}$

Step 1.2.8.1.5

Pull terms out from under the radical.

$x = \frac{- 26 \pm 6 \sqrt{33}}{2 \cdot - 1}$

Step 1.2.8.2

Multiply $2$ by $- 1$ .

$x = \frac{- 26 \pm 6 \sqrt{33}}{- 2}$

Step 1.2.8.3

Simplify $\frac{- 26 \pm 6 \sqrt{33}}{- 2}$ .

$x = 13 \pm 3 \sqrt{33}$

Step 1.2.8.4

Change the $\pm$ to $-$ .

$x = 13 - 3 \sqrt{33}$

Step 1.2.9

The final answer is the combination of both solutions.

$x = 13 + 3 \sqrt{33}, 13 - 3 \sqrt{33}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(13 + 3 \sqrt{33}, 0), (13 - 3 \sqrt{33}, 0)$

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 $y$ .

$y = - \frac{1}{64} \cdot {(0)}^{2} + \frac{13}{32} \cdot (0) + 2$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - \frac{1}{64} \cdot 0^{2} + \frac{13}{32} \cdot (0) + 2$

Step 2.2.2

Multiply $\frac{13}{32}$ by $0$ .

$y = - \frac{1}{64} \cdot 0^{2} + \frac{13}{32} \cdot 0 + 2$

Step 2.2.3

Remove parentheses.

$y = - \frac{1}{64} \cdot {(0)}^{2} + \frac{13}{32} \cdot (0) + 2$

Step 2.2.4

Simplify $- \frac{1}{64} \cdot {(0)}^{2} + \frac{13}{32} \cdot (0) + 2$ .

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

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

$y = - \frac{1}{64} \cdot 0 + \frac{13}{32} \cdot (0) + 2$

Step 2.2.4.1.2

Multiply $- \frac{1}{64} \cdot 0$ .

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Step 2.2.4.1.2.1

Multiply $0$ by $- 1$ .

$y = 0 (\frac{1}{64}) + \frac{13}{32} \cdot (0) + 2$

Step 2.2.4.1.2.2

Multiply $0$ by $\frac{1}{64}$ .

$y = 0 + \frac{13}{32} \cdot (0) + 2$

Step 2.2.4.1.3

Multiply $\frac{13}{32}$ by $0$ .

$y = 0 + 0 + 2$

Step 2.2.4.2

Simplify by adding numbers.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 2$

Step 2.2.4.2.2

Add $0$ and $2$ .

$y = 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

x-intercept(s): $(13 + 3 \sqrt{33}, 0), (13 - 3 \sqrt{33}, 0)$

y-intercept(s): $(0, 2)$

Step 4