Algebra Examples

$y = - x^{2} + 3 x + 13$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $- x^{2} + 3 x + 13$ .

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Step 1.1.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 1$

$b = 3$

$c = 13$

Step 1.1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{3}{2 \cdot - 1}$

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Multiply $2$ by $- 1$ .

$d = \frac{3}{- 2}$

Step 1.1.3.2.2

Move the negative in front of the fraction.

$d = - \frac{3}{2}$

Step 1.1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.1.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 13 - \frac{3^{2}}{4 \cdot - 1}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

Raise $3$ to the power of $2$ .

$e = 13 - \frac{9}{4 \cdot - 1}$

Step 1.1.4.2.1.2

Multiply $4$ by $- 1$ .

$e = 13 - \frac{9}{- 4}$

Step 1.1.4.2.1.3

Move the negative in front of the fraction.

$e = 13 - - \frac{9}{4}$

Step 1.1.4.2.1.4

Multiply $- - \frac{9}{4}$ .

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Step 1.1.4.2.1.4.1

Multiply $- 1$ by $- 1$ .

$e = 13 + 1 (\frac{9}{4})$

Step 1.1.4.2.1.4.2

Multiply $\frac{9}{4}$ by $1$ .

$e = 13 + \frac{9}{4}$

Step 1.1.4.2.2

To write $13$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 13 \cdot \frac{4}{4} + \frac{9}{4}$

Step 1.1.4.2.3

Combine $13$ and $\frac{4}{4}$ .

$e = \frac{13 \cdot 4}{4} + \frac{9}{4}$

Step 1.1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{13 \cdot 4 + 9}{4}$

Step 1.1.4.2.5

Simplify the numerator.

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Step 1.1.4.2.5.1

Multiply $13$ by $4$ .

$e = \frac{52 + 9}{4}$

Step 1.1.4.2.5.2

Add $52$ and $9$ .

$e = \frac{61}{4}$

Step 1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - \frac{3}{2})}^{2} + \frac{61}{4}$ .

$- {(x - \frac{3}{2})}^{2} + \frac{61}{4}$

Step 1.2

Set $y$ equal to the new right side.

$y = - {(x - \frac{3}{2})}^{2} + \frac{61}{4}$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - 1$

$h = \frac{3}{2}$

$k = \frac{61}{4}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{3}{2}, \frac{61}{4})$

Step 4

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