Algebra Examples

$y = \frac{3}{5} x^{2} + 30 x + 382$

Step 1

Combine $\frac{3}{5}$ and $x^{2}$ .

$y = \frac{3 x^{2}}{5} + 30 x + 382$

Step 2

Complete the square for $\frac{3 x^{2}}{5} + 30 x + 382$ .

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Step 2.1

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

$a = \frac{3}{5}$

$b = 30$

$c = 382$

Step 2.2

Consider the vertex form of a parabola.

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

Step 2.3

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

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Step 2.3.1

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

$d = \frac{30}{2 (\frac{3}{5})}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $30$ and $2$ .

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Step 2.3.2.1.1

Factor $2$ out of $30$ .

$d = \frac{2 \cdot 15}{2 (\frac{3}{5})}$

Step 2.3.2.1.2

Cancel the common factors.

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Step 2.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot 15}{2 (\frac{3}{5})}$

Step 2.3.2.1.2.2

Rewrite the expression.

$d = \frac{15}{\frac{3}{5}}$

Step 2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 15 (\frac{5}{3})$

Step 2.3.2.3

Cancel the common factor of $3$ .

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Step 2.3.2.3.1

Factor $3$ out of $15$ .

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

Step 2.3.2.3.2

Cancel the common factor.

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

Step 2.3.2.3.3

Rewrite the expression.

$d = 5 \cdot 5$

Step 2.3.2.4

Multiply $5$ by $5$ .

$d = 25$

Step 2.4

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

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Step 2.4.1

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

$e = 382 - \frac{30^{2}}{4 (\frac{3}{5})}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Raise $30$ to the power of $2$ .

$e = 382 - \frac{900}{4 (\frac{3}{5})}$

Step 2.4.2.1.2

Combine $4$ and $\frac{3}{5}$ .

$e = 382 - \frac{900}{\frac{4 \cdot 3}{5}}$

Step 2.4.2.1.3

Multiply $4$ by $3$ .

$e = 382 - \frac{900}{\frac{12}{5}}$

Step 2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = 382 - (900 (\frac{5}{12}))$

Step 2.4.2.1.5

Cancel the common factor of $12$ .

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Step 2.4.2.1.5.1

Factor $12$ out of $900$ .

$e = 382 - (12 (75) \frac{5}{12})$

Step 2.4.2.1.5.2

Cancel the common factor.

$e = 382 - (12 \cdot 75 \frac{5}{12})$

Step 2.4.2.1.5.3

Rewrite the expression.

$e = 382 - (75 \cdot 5)$

Step 2.4.2.1.6

Multiply $75$ by $5$ .

$e = 382 - 1 \cdot 375$

Step 2.4.2.1.7

Multiply $- 1$ by $375$ .

$e = 382 - 375$

Step 2.4.2.2

Subtract $375$ from $382$ .

$e = 7$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{3}{5} {(x + 25)}^{2} + 7$ .

$\frac{3}{5} {(x + 25)}^{2} + 7$

Step 3

Set $y$ equal to the new right side.

$y = \frac{3}{5} \cdot {(x + 25)}^{2} + 7$