Algebra Examples

$y = 0.5 {(x - 3)}^{2}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $0.5 {(x - 3)}^{2}$ .

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Step 1.1.1

Simplify the expression.

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Step 1.1.1.1

Rewrite ${(x - 3)}^{2}$ as $(x - 3) (x - 3)$ .

$0.5 ((x - 3) (x - 3))$

Step 1.1.1.2

Expand $(x - 3) (x - 3)$ using the FOIL Method.

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Step 1.1.1.2.1

Apply the distributive property.

$0.5 (x (x - 3) - 3 (x - 3))$

Step 1.1.1.2.2

Apply the distributive property.

$0.5 (x \cdot x + x \cdot - 3 - 3 (x - 3))$

Step 1.1.1.2.3

Apply the distributive property.

$0.5 (x \cdot x + x \cdot - 3 - 3 x - 3 \cdot - 3)$

Step 1.1.1.3

Simplify and combine like terms.

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Step 1.1.1.3.1

Simplify each term.

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Step 1.1.1.3.1.1

Multiply $x$ by $x$ .

$0.5 (x^{2} + x \cdot - 3 - 3 x - 3 \cdot - 3)$

Step 1.1.1.3.1.2

Move $- 3$ to the left of $x$ .

$0.5 (x^{2} - 3 \cdot x - 3 x - 3 \cdot - 3)$

Step 1.1.1.3.1.3

Multiply $- 3$ by $- 3$ .

$0.5 (x^{2} - 3 x - 3 x + 9)$

Step 1.1.1.3.2

Subtract $3 x$ from $- 3 x$ .

$0.5 (x^{2} - 6 x + 9)$

Step 1.1.1.4

Apply the distributive property.

$0.5 x^{2} + 0.5 (- 6 x) + 0.5 \cdot 9$

Step 1.1.1.5

Simplify.

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Step 1.1.1.5.1

Multiply $- 6$ by $0.5$ .

$0.5 x^{2} - 3 x + 0.5 \cdot 9$

Step 1.1.1.5.2

Multiply $0.5$ by $9$ .

$0.5 x^{2} - 3 x + 4.5$

Step 1.1.2

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

$a = 0.5$

$b = - 3$

$c = 4.5$

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

$d = \frac{- 3}{2 \cdot 0.5}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Cancel the common factor of $- 3$ and $2$ .

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Step 1.1.4.2.1.1

Rewrite $- 3$ as $1 (- 3)$ .

$d = \frac{1 (- 3)}{2 \cdot 0.5}$

Step 1.1.4.2.1.2

Cancel the common factors.

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Step 1.1.4.2.1.2.1

Factor $1$ out of $2 \cdot 0.5$ .

$d = \frac{1 (- 3)}{1 (2 \cdot 0.5)}$

Step 1.1.4.2.1.2.2

Cancel the common factor.

$d = \frac{1 \cdot - 3}{1 (2 \cdot 0.5)}$

Step 1.1.4.2.1.2.3

Rewrite the expression.

$d = \frac{- 3}{2 \cdot 0.5}$

Step 1.1.4.2.2

Multiply $2$ by $0.5$ .

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

Step 1.1.4.2.3

Divide $- 3$ by $1$ .

$d = - 3$

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

$e = 4.5 - \frac{{(- 3)}^{2}}{4 \cdot 0.5}$

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

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

$e = 4.5 - \frac{9}{4 \cdot 0.5}$

Step 1.1.5.2.1.2

Multiply $4$ by $0.5$ .

$e = 4.5 - \frac{9}{2}$

Step 1.1.5.2.1.3

Cancel the common factor of $9$ and $2$ .

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Step 1.1.5.2.1.3.1

Rewrite $9$ as $1 (9)$ .

$e = 4.5 - \frac{1 (9)}{2}$

Step 1.1.5.2.1.3.2

Cancel the common factors.

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Step 1.1.5.2.1.3.2.1

Rewrite $2$ as $1 (2)$ .

$e = 4.5 - \frac{1 \cdot 9}{1 \cdot 2}$

Step 1.1.5.2.1.3.2.2

Cancel the common factor.

$e = 4.5 - \frac{1 \cdot 9}{1 \cdot 2}$

Step 1.1.5.2.1.3.2.3

Rewrite the expression.

$e = 4.5 - \frac{9}{2}$

Step 1.1.5.2.2

To write $4.5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = 4.5 \cdot \frac{2}{2} - \frac{9}{2}$

Step 1.1.5.2.3

Combine $4.5$ and $\frac{2}{2}$ .

$e = \frac{4.5 \cdot 2}{2} - \frac{9}{2}$

Step 1.1.5.2.4

Combine the numerators over the common denominator.

$e = \frac{4.5 \cdot 2 - 9}{2}$

Step 1.1.5.2.5

Simplify the numerator.

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Step 1.1.5.2.5.1

Multiply $4.5$ by $2$ .

$e = \frac{9 - 9}{2}$

Step 1.1.5.2.5.2

Subtract $9$ from $9$ .

$e = \frac{0}{2}$

Step 1.1.5.2.6

Divide $0$ by $2$ .

$e = 0$

Step 1.1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $0.5 {(x - 3)}^{2} + 0$ .

$0.5 {(x - 3)}^{2} + 0$

Step 1.2

Set $y$ equal to the new right side.

$y = 0.5 {(x - 3)}^{2} + 0$

Step 2

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

$a = 0.5$

$h = 3$

$k = 0$

Step 3

Find the vertex $(h, k)$ .

$(3, 0)$

Step 4