Examples

f (x) = 7 - 3 x + 2 x^{2}

$f (x) = 7 - 3 x + 2 x^{2}$

Step 1

Write

f (x) = 7 - 3 x + 2 x^{2}

$f (x) = 7 - 3 x + 2 x^{2}$ as an equation.

y = 7 - 3 x + 2 x^{2}

$y = 7 - 3 x + 2 x^{2}$

Step 2

Since

x

$x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

7 - 3 x + 2 x^{2} = y

$7 - 3 x + 2 x^{2} = y$

Step 3

Subtract

7

$7$ from both sides of the equation.

- 3 x + 2 x^{2} = y - 7

$- 3 x + 2 x^{2} = y - 7$

Step 4

Complete the square.

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Step 4.1

Reorder

- 3 x

$- 3 x$ and

2 x^{2}

$2 x^{2}$ .

2 x^{2} - 3 x = y - 7

$2 x^{2} - 3 x = y - 7$

Step 4.2

Use the form

a x^{2} + b x + c

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

a

$a$ ,

b

$b$ , and

c

$c$ .

a = 2

$a = 2$

b = - 3

$b = - 3$

c = 0 = y - 7

$c = 0 = y - 7$

Step 4.3

Consider the vertex form of a parabola.

a {(x + d)}^{2} + e = y - 7

$a {(x + d)}^{2} + e = y - 7$

Step 4.4

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

$d = \frac{b}{2 a}$ .

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Step 4.4.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

$d = \frac{b}{2 a}$ .

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

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

Step 4.4.2

Simplify the right side.

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Step 4.4.2.1

Multiply

2

$2$ by

2

$2$ .

d = \frac{- 3}{4}

$d = \frac{- 3}{4}$

Step 4.4.2.2

Move the negative in front of the fraction.

d = - \frac{3}{4}

$d = - \frac{3}{4}$

d = - \frac{3}{4}

$d = - \frac{3}{4}$

d = - \frac{3}{4}

$d = - \frac{3}{4}$

Step 4.5

Find the value of

e

$e$ using the formula

e = c - \frac{b^{2}}{4 a}

$e = c - \frac{b^{2}}{4 a}$ .

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Step 4.5.1

Substitute the values of

c

$c$ ,

b

$b$ and

a

$a$ into the formula

e = c - \frac{b^{2}}{4 a}

$e = c - \frac{b^{2}}{4 a}$ .

e = 0 - \frac{{(- 3)}^{2}}{4 \cdot 2}

$e = 0 - \frac{{(- 3)}^{2}}{4 \cdot 2}$

Step 4.5.2

Simplify the right side.

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Step 4.5.2.1

Simplify each term.

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Step 4.5.2.1.1

Raise

- 3

$- 3$ to the power of

2

$2$ .

e = 0 - \frac{9}{4 \cdot 2}

$e = 0 - \frac{9}{4 \cdot 2}$

Step 4.5.2.1.2

Multiply

4

$4$ by

2

$2$ .

e = 0 - \frac{9}{8}

$e = 0 - \frac{9}{8}$

e = 0 - \frac{9}{8}

$e = 0 - \frac{9}{8}$

Step 4.5.2.2

Subtract

\frac{9}{8}

$\frac{9}{8}$ from

0

$0$ .

e = - \frac{9}{8}

$e = - \frac{9}{8}$

e = - \frac{9}{8}

$e = - \frac{9}{8}$

e = - \frac{9}{8}

$e = - \frac{9}{8}$

Step 4.6

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

2 {(x - \frac{3}{4})}^{2} - \frac{9}{8}

$2 {(x - \frac{3}{4})}^{2} - \frac{9}{8}$ .

2 {(x - \frac{3}{4})}^{2} - \frac{9}{8} = y - 7

$2 {(x - \frac{3}{4})}^{2} - \frac{9}{8} = y - 7$

2 {(x - \frac{3}{4})}^{2} - \frac{9}{8} = y - 7

$2 {(x - \frac{3}{4})}^{2} - \frac{9}{8} = y - 7$

Step 5

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 5.1

Add

\frac{9}{8}

$\frac{9}{8}$ to both sides of the equation.

2 {(x - \frac{3}{4})}^{2} = y - 7 + \frac{9}{8}

$2 {(x - \frac{3}{4})}^{2} = y - 7 + \frac{9}{8}$

Step 5.2

To write

- 7

$- 7$ as a fraction with a common denominator, multiply by

\frac{8}{8}

$\frac{8}{8}$ .

2 {(x - \frac{3}{4})}^{2} = y - 7 \cdot \frac{8}{8} + \frac{9}{8}

$2 {(x - \frac{3}{4})}^{2} = y - 7 \cdot \frac{8}{8} + \frac{9}{8}$

Step 5.3

Combine

- 7

$- 7$ and

\frac{8}{8}

$\frac{8}{8}$ .

2 {(x - \frac{3}{4})}^{2} = y + \frac{- 7 \cdot 8}{8} + \frac{9}{8}

$2 {(x - \frac{3}{4})}^{2} = y + \frac{- 7 \cdot 8}{8} + \frac{9}{8}$

Step 5.4

Combine the numerators over the common denominator.

2 {(x - \frac{3}{4})}^{2} = y + \frac{- 7 \cdot 8 + 9}{8}

$2 {(x - \frac{3}{4})}^{2} = y + \frac{- 7 \cdot 8 + 9}{8}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Multiply

- 7

$- 7$ by

8

$8$ .

2 {(x - \frac{3}{4})}^{2} = y + \frac{- 56 + 9}{8}

$2 {(x - \frac{3}{4})}^{2} = y + \frac{- 56 + 9}{8}$

Step 5.5.2

Add

- 56

$- 56$ and

9

$9$ .

2 {(x - \frac{3}{4})}^{2} = y + \frac{- 47}{8}

$2 {(x - \frac{3}{4})}^{2} = y + \frac{- 47}{8}$

2 {(x - \frac{3}{4})}^{2} = y + \frac{- 47}{8}

$2 {(x - \frac{3}{4})}^{2} = y + \frac{- 47}{8}$

Step 5.6

Move the negative in front of the fraction.

2 {(x - \frac{3}{4})}^{2} = y - \frac{47}{8}

$2 {(x - \frac{3}{4})}^{2} = y - \frac{47}{8}$

2 {(x - \frac{3}{4})}^{2} = y - \frac{47}{8}

$2 {(x - \frac{3}{4})}^{2} = y - \frac{47}{8}$

Step 6

Divide each term in

2 {(x - \frac{3}{4})}^{2} = y - \frac{47}{8}

$2 {(x - \frac{3}{4})}^{2} = y - \frac{47}{8}$ by

2

$2$ and simplify.

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Step 6.1

Divide each term in

2 {(x - \frac{3}{4})}^{2} = y - \frac{47}{8}

$2 {(x - \frac{3}{4})}^{2} = y - \frac{47}{8}$ by

2

$2$ .

\frac{2 {(x - \frac{3}{4})}^{2}}{2} = \frac{y}{2} + \frac{- \frac{47}{8}}{2}

$\frac{2 {(x - \frac{3}{4})}^{2}}{2} = \frac{y}{2} + \frac{- \frac{47}{8}}{2}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of

2

$2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{2 {(x - \frac{3}{4})}^{2}}{2} = \frac{y}{2} + \frac{- \frac{47}{8}}{2}$

Step 6.2.1.2

Divide

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

$1$ .

${(x - \frac{3}{4})}^{2} = \frac{y}{2} + \frac{- \frac{47}{8}}{2}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Multiply the numerator by the reciprocal of the denominator.

${(x - \frac{3}{4})}^{2} = \frac{y}{2} - \frac{47}{8} \cdot \frac{1}{2}$

Step 6.3.1.2

Multiply

$- \frac{47}{8} \cdot \frac{1}{2}$ .

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Step 6.3.1.2.1

Multiply

$\frac{1}{2}$ by

$\frac{47}{8}$ .

${(x - \frac{3}{4})}^{2} = \frac{y}{2} - \frac{47}{2 \cdot 8}$

Step 6.3.1.2.2

Multiply

$2$ by

$8$ .

${(x - \frac{3}{4})}^{2} = \frac{y}{2} - \frac{47}{16}$

Step 7

Factor.

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Step 7.1

To write

$\frac{y}{2}$ as a fraction with a common denominator, multiply by

$\frac{8}{8}$ .

${(x - \frac{3}{4})}^{2} = \frac{y}{2} \cdot \frac{8}{8} - \frac{47}{16}$

Step 7.2

Write each expression with a common denominator of

$16$ , by multiplying each by an appropriate factor of

$1$ .

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Step 7.2.1

Multiply

$\frac{y}{2}$ by

$\frac{8}{8}$ .

${(x - \frac{3}{4})}^{2} = \frac{y \cdot 8}{2 \cdot 8} - \frac{47}{16}$

Step 7.2.2

Multiply

$2$ by

$8$ .

${(x - \frac{3}{4})}^{2} = \frac{y \cdot 8}{16} - \frac{47}{16}$

Step 7.3

Combine the numerators over the common denominator.

${(x - \frac{3}{4})}^{2} = \frac{y \cdot 8 - 47}{16}$

Step 7.4

Move

$8$ to the left of

$y$ .

${(x - \frac{3}{4})}^{2} = \frac{8 y - 47}{16}$

Step 7.5

Reorder terms.

${(x - \frac{3}{4})}^{2} = \frac{1}{16} (8 y - 47)$

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