Algebra Examples

$d (x) = \frac{1}{2} x^{2} - 10 x$

Step 1

Write $d (x) = \frac{1}{2} x^{2} - 10 x$ as an equation.

$y = \frac{1}{2} x^{2} - 10 x$

Step 2

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

$\frac{1}{2} x^{2} - 10 x = y$

Step 3

Complete the square.

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Step 3.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{x^{2}}{2} - 10 x = y$

Step 3.2

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

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

$b = - 10$

$c = 0 = y$

Step 3.3

Consider the vertex form of a parabola.

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

Step 3.4

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

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Step 3.4.1

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

$d = \frac{- 10}{2 (\frac{1}{2})}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Cancel the common factor of $- 10$ and $2$ .

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Step 3.4.2.1.1

Factor $2$ out of $- 10$ .

$d = \frac{2 \cdot - 5}{2 (\frac{1}{2})}$

Step 3.4.2.1.2

Cancel the common factors.

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Step 3.4.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot - 5}{2 (\frac{1}{2})}$

Step 3.4.2.1.2.2

Rewrite the expression.

$d = \frac{- 5}{\frac{1}{2}}$

Step 3.4.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - 5 \cdot 2$

Step 3.4.2.3

Multiply $- 5$ by $2$ .

$d = - 10$

Step 3.5

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

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Step 3.5.1

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

$e = 0 - \frac{{(- 10)}^{2}}{4 (\frac{1}{2})}$

Step 3.5.2

Simplify the right side.

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Step 3.5.2.1

Simplify each term.

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Step 3.5.2.1.1

Raise $- 10$ to the power of $2$ .

$e = 0 - \frac{100}{4 (\frac{1}{2})}$

Step 3.5.2.1.2

Combine $4$ and $\frac{1}{2}$ .

$e = 0 - \frac{100}{\frac{4}{2}}$

Step 3.5.2.1.3

Divide $4$ by $2$ .

$e = 0 - \frac{100}{2}$

Step 3.5.2.1.4

Divide $100$ by $2$ .

$e = 0 - 1 \cdot 50$

Step 3.5.2.1.5

Multiply $- 1$ by $50$ .

$e = 0 - 50$

Step 3.5.2.2

Subtract $50$ from $0$ .

$e = - 50$

Step 3.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{2} {(x - 10)}^{2} - 50$ .

$\frac{1}{2} \cdot {(x - 10)}^{2} - 50 = y$

Step 4

Add $50$ to both sides of the equation.

$\frac{1}{2} \cdot {(x - 10)}^{2} = y + 50$

Step 5

Multiply each term in $\frac{1}{2} \cdot {(x - 10)}^{2} = y + 50$ by $2$ to eliminate the fractions.

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Step 5.1

Multiply each term in $\frac{1}{2} \cdot {(x - 10)}^{2} = y + 50$ by $2$ .

$\frac{1}{2} \cdot {(x - 10)}^{2} \cdot 2 = y \cdot 2 + 50 \cdot 2$

Step 5.2

Simplify the left side.

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Step 5.2.1

Combine $\frac{1}{2}$ and ${(x - 10)}^{2}$ .

$\frac{{(x - 10)}^{2}}{2} \cdot 2 = y \cdot 2 + 50 \cdot 2$

Step 5.2.2

Cancel the common factor of $2$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{{(x - 10)}^{2}}{2} \cdot 2 = y \cdot 2 + 50 \cdot 2$

Step 5.2.2.2

Rewrite the expression.

${(x - 10)}^{2} = y \cdot 2 + 50 \cdot 2$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Move $2$ to the left of $y$ .

${(x - 10)}^{2} = 2 \cdot y + 50 \cdot 2$

Step 5.3.1.2

Multiply $50$ by $2$ .

${(x - 10)}^{2} = 2 y + 100$

Step 6

Factor $2$ out of $2 y + 100$ .

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Step 6.1

Factor $2$ out of $2 y$ .

${(x - 10)}^{2} = 2 (y) + 100$

Step 6.2

Factor $2$ out of $100$ .

${(x - 10)}^{2} = 2 y + 2 \cdot 50$

Step 6.3

Factor $2$ out of $2 y + 2 \cdot 50$ .

${(x - 10)}^{2} = 2 (y + 50)$