Algebra Examples

$y = - {(x - 0.5)}^{2} + 9$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $- {(x - 0.5)}^{2} + 9$ .

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Step 1.1.1

Simplify the expression.

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Step 1.1.1.1

Simplify each term.

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Step 1.1.1.1.1

Rewrite ${(x - 0.5)}^{2}$ as $(x - 0.5) (x - 0.5)$ .

$- ((x - 0.5) (x - 0.5)) + 9$

Step 1.1.1.1.2

Expand $(x - 0.5) (x - 0.5)$ using the FOIL Method.

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Step 1.1.1.1.2.1

Apply the distributive property.

$- (x (x - 0.5) - 0.5 (x - 0.5)) + 9$

Step 1.1.1.1.2.2

Apply the distributive property.

$- (x \cdot x + x \cdot - 0.5 - 0.5 (x - 0.5)) + 9$

Step 1.1.1.1.2.3

Apply the distributive property.

$- (x \cdot x + x \cdot - 0.5 - 0.5 x - 0.5 \cdot - 0.5) + 9$

Step 1.1.1.1.3

Simplify and combine like terms.

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Step 1.1.1.1.3.1

Simplify each term.

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Step 1.1.1.1.3.1.1

Multiply $x$ by $x$ .

$- (x^{2} + x \cdot - 0.5 - 0.5 x - 0.5 \cdot - 0.5) + 9$

Step 1.1.1.1.3.1.2

Move $- 0.5$ to the left of $x$ .

$- (x^{2} - 0.5 \cdot x - 0.5 x - 0.5 \cdot - 0.5) + 9$

Step 1.1.1.1.3.1.3

Multiply $- 0.5$ by $- 0.5$ .

$- (x^{2} - 0.5 x - 0.5 x + 0.25) + 9$

Step 1.1.1.1.3.2

Subtract $0.5 x$ from $- 0.5 x$ .

$- (x^{2} - 1 x + 0.25) + 9$

Step 1.1.1.1.4

Apply the distributive property.

$- x^{2} - (- 1 x) - 1 \cdot 0.25 + 9$

Step 1.1.1.1.5

Simplify.

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Step 1.1.1.1.5.1

Multiply $- (- 1 x)$ .

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Step 1.1.1.1.5.1.1

Multiply $- 1$ by $- 1$ .

$- x^{2} + 1 x - 1 \cdot 0.25 + 9$

Step 1.1.1.1.5.1.2

Multiply $x$ by $1$ .

$- x^{2} + x - 1 \cdot 0.25 + 9$

Step 1.1.1.1.5.2

Multiply $- 1$ by $0.25$ .

$- x^{2} + x - 0.25 + 9$

Step 1.1.1.2

Add $- 0.25$ and $9$ .

$- x^{2} + x + 8.75$

Step 1.1.2

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

$a = - 1$

$b = 1$

$c = 8.75$

Step 1.1.3

Consider the vertex form of a parabola.

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

Step 1.1.4

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

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Step 1.1.4.1

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

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

Step 1.1.4.2

Cancel the common factor of $1$ and $- 1$ .

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Step 1.1.4.2.1

Rewrite $1$ as $- 1 (- 1)$ .

$d = \frac{- 1 (- 1)}{2 \cdot - 1}$

Step 1.1.4.2.2

Move the negative in front of the fraction.

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

Step 1.1.5

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

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Step 1.1.5.1

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

$e = 8.75 - \frac{1^{2}}{4 \cdot - 1}$

Step 1.1.5.2

Simplify the right side.

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Step 1.1.5.2.1

Simplify each term.

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Step 1.1.5.2.1.1

One to any power is one.

$e = 8.75 - \frac{1}{4 \cdot - 1}$

Step 1.1.5.2.1.2

Multiply $4$ by $- 1$ .

$e = 8.75 - \frac{1}{- 4}$

Step 1.1.5.2.1.3

Move the negative in front of the fraction.

$e = 8.75 - - \frac{1}{4}$

Step 1.1.5.2.1.4

Multiply $- - \frac{1}{4}$ .

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Step 1.1.5.2.1.4.1

Multiply $- 1$ by $- 1$ .

$e = 8.75 + 1 (\frac{1}{4})$

Step 1.1.5.2.1.4.2

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

$e = 8.75 + \frac{1}{4}$

Step 1.1.5.2.2

To write $8.75$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 8.75 \cdot \frac{4}{4} + \frac{1}{4}$

Step 1.1.5.2.3

Combine $8.75$ and $\frac{4}{4}$ .

$e = \frac{8.75 \cdot 4}{4} + \frac{1}{4}$

Step 1.1.5.2.4

Combine the numerators over the common denominator.

$e = \frac{8.75 \cdot 4 + 1}{4}$

Step 1.1.5.2.5

Simplify the numerator.

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Step 1.1.5.2.5.1

Multiply $8.75$ by $4$ .

$e = \frac{35 + 1}{4}$

Step 1.1.5.2.5.2

Add $35$ and $1$ .

$e = \frac{36}{4}$

Step 1.1.5.2.6

Divide $36$ by $4$ .

$e = 9$

Step 1.1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - \frac{1}{2})}^{2} + 9$ .

$- {(x - \frac{1}{2})}^{2} + 9$

Step 1.2

Set $y$ equal to the new right side.

$y = - {(x - \frac{1}{2})}^{2} + 9$

Step 2

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

$a = - 1$

$h = \frac{1}{2}$

$k = 9$

Step 3

Find the vertex $(h, k)$ .

$(\frac{1}{2}, 9)$

Step 4