Precalculus Examples

$y = 5 x^{2} + 2 x - 5$

Step 1

Complete the square for $5 x^{2} + 2 x - 5$ .

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Step 1.1

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

$a = 5$

$b = 2$

$c = - 5$

Step 1.2

Consider the vertex form of a parabola.

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

Step 1.3

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

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Step 1.3.1

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

$d = \frac{2}{2 \cdot 5}$

Step 1.3.2

Cancel the common factor of $2$ .

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Step 1.3.2.1

Cancel the common factor.

$d = \frac{2}{2 \cdot 5}$

Step 1.3.2.2

Rewrite the expression.

$d = \frac{1}{5}$

Step 1.4

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

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Step 1.4.1

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

$e = - 5 - \frac{2^{2}}{4 \cdot 5}$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Simplify each term.

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Step 1.4.2.1.1

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

$e = - 5 - \frac{4}{4 \cdot 5}$

Step 1.4.2.1.2

Multiply $4$ by $5$ .

$e = - 5 - \frac{4}{20}$

Step 1.4.2.1.3

Cancel the common factor of $4$ and $20$ .

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Step 1.4.2.1.3.1

Factor $4$ out of $4$ .

$e = - 5 - \frac{4 (1)}{20}$

Step 1.4.2.1.3.2

Cancel the common factors.

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Step 1.4.2.1.3.2.1

Factor $4$ out of $20$ .

$e = - 5 - \frac{4 \cdot 1}{4 \cdot 5}$

Step 1.4.2.1.3.2.2

Cancel the common factor.

$e = - 5 - \frac{4 \cdot 1}{4 \cdot 5}$

Step 1.4.2.1.3.2.3

Rewrite the expression.

$e = - 5 - \frac{1}{5}$

Step 1.4.2.2

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$e = - 5 \cdot \frac{5}{5} - \frac{1}{5}$

Step 1.4.2.3

Combine $- 5$ and $\frac{5}{5}$ .

$e = \frac{- 5 \cdot 5}{5} - \frac{1}{5}$

Step 1.4.2.4

Combine the numerators over the common denominator.

$e = \frac{- 5 \cdot 5 - 1}{5}$

Step 1.4.2.5

Simplify the numerator.

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Step 1.4.2.5.1

Multiply $- 5$ by $5$ .

$e = \frac{- 25 - 1}{5}$

Step 1.4.2.5.2

Subtract $1$ from $- 25$ .

$e = \frac{- 26}{5}$

Step 1.4.2.6

Move the negative in front of the fraction.

$e = - \frac{26}{5}$

Step 1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $5 {(x + \frac{1}{5})}^{2} - \frac{26}{5}$ .

$5 {(x + \frac{1}{5})}^{2} - \frac{26}{5}$

Step 2

Set $y$ equal to the new right side.

$y = 5 {(x + \frac{1}{5})}^{2} - \frac{26}{5}$

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