Algebra Examples

$- \frac{2}{5} x^{2} + 6 x$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $x^{2}$ and $\frac{2}{5}$ .

$y = - \frac{x^{2} \cdot 2}{5} + 6 x$

Step 1.1.2

Move $2$ to the left of $x^{2}$ .

$y = - \frac{2 x^{2}}{5} + 6 x$

Step 1.2

Complete the square for $- \frac{2 x^{2}}{5} + 6 x$ .

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Step 1.2.1

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

$a = - \frac{2}{5}$

$b = 6$

$c = 0$

Step 1.2.2

Consider the vertex form of a parabola.

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

Step 1.2.3

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

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Step 1.2.3.1

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

$d = \frac{6}{2 (- \frac{2}{5})}$

Step 1.2.3.2

Simplify the right side.

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Step 1.2.3.2.1

Cancel the common factor of $6$ and $2$ .

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Step 1.2.3.2.1.1

Factor $2$ out of $6$ .

$d = \frac{2 \cdot 3}{2 (- \frac{2}{5})}$

Step 1.2.3.2.1.2

Cancel the common factors.

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Step 1.2.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot 3}{2 (- \frac{2}{5})}$

Step 1.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{3}{- \frac{2}{5}}$

Step 1.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 3 (- \frac{5}{2})$

Step 1.2.3.2.3

Multiply $3 (- \frac{5}{2})$ .

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Step 1.2.3.2.3.1

Multiply $- 1$ by $3$ .

$d = - 3 (\frac{5}{2})$

Step 1.2.3.2.3.2

Combine $- 3$ and $\frac{5}{2}$ .

$d = \frac{- 3 \cdot 5}{2}$

Step 1.2.3.2.3.3

Multiply $- 3$ by $5$ .

$d = \frac{- 15}{2}$

Step 1.2.3.2.4

Move the negative in front of the fraction.

$d = - \frac{15}{2}$

Step 1.2.4

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

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Step 1.2.4.1

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

$e = 0 - \frac{6^{2}}{4 (- \frac{2}{5})}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify each term.

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Step 1.2.4.2.1.1

Raise $6$ to the power of $2$ .

$e = 0 - \frac{36}{4 (- \frac{2}{5})}$

Step 1.2.4.2.1.2

Simplify the denominator.

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Step 1.2.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 0 - \frac{36}{- 4 (\frac{2}{5})}$

Step 1.2.4.2.1.2.2

Combine $- 4$ and $\frac{2}{5}$ .

$e = 0 - \frac{36}{\frac{- 4 \cdot 2}{5}}$

Step 1.2.4.2.1.3

Multiply $- 4$ by $2$ .

$e = 0 - \frac{36}{\frac{- 8}{5}}$

Step 1.2.4.2.1.4

Move the negative in front of the fraction.

$e = 0 - \frac{36}{- \frac{8}{5}}$

Step 1.2.4.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (36 (- \frac{5}{8}))$

Step 1.2.4.2.1.6

Cancel the common factor of $4$ .

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Step 1.2.4.2.1.6.1

Move the leading negative in $- \frac{5}{8}$ into the numerator.

$e = 0 - (36 (\frac{- 5}{8}))$

Step 1.2.4.2.1.6.2

Factor $4$ out of $36$ .

$e = 0 - (4 (9) \frac{- 5}{8})$

Step 1.2.4.2.1.6.3

Factor $4$ out of $8$ .

$e = 0 - (4 \cdot 9 \frac{- 5}{4 \cdot 2})$

Step 1.2.4.2.1.6.4

Cancel the common factor.

$e = 0 - (4 \cdot 9 \frac{- 5}{4 \cdot 2})$

Step 1.2.4.2.1.6.5

Rewrite the expression.

$e = 0 - (9 (\frac{- 5}{2}))$

Step 1.2.4.2.1.7

Combine $9$ and $\frac{- 5}{2}$ .

$e = 0 - \frac{9 \cdot - 5}{2}$

Step 1.2.4.2.1.8

Multiply $9$ by $- 5$ .

$e = 0 - \frac{- 45}{2}$

Step 1.2.4.2.1.9

Move the negative in front of the fraction.

$e = 0 - - \frac{45}{2}$

Step 1.2.4.2.1.10

Multiply $- - \frac{45}{2}$ .

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Step 1.2.4.2.1.10.1

Multiply $- 1$ by $- 1$ .

$e = 0 + 1 (\frac{45}{2})$

Step 1.2.4.2.1.10.2

Multiply $\frac{45}{2}$ by $1$ .

$e = 0 + \frac{45}{2}$

Step 1.2.4.2.2

Add $0$ and $\frac{45}{2}$ .

$e = \frac{45}{2}$

Step 1.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{2}{5} {(x - \frac{15}{2})}^{2} + \frac{45}{2}$ .

$- \frac{2}{5} {(x - \frac{15}{2})}^{2} + \frac{45}{2}$

Step 1.3

Set $y$ equal to the new right side.

$y = - \frac{2}{5} \cdot {(x - \frac{15}{2})}^{2} + \frac{45}{2}$

Step 2

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

$a = - \frac{2}{5}$

$h = \frac{15}{2}$

$k = \frac{45}{2}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{15}{2}, \frac{45}{2})$

Step 4