Precalculus Examples

$10 + 5 x = - {(1 - y)}^{2}$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Isolate $x$ to the left side of the equation.

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Step 1.1.1

Subtract $10$ from both sides of the equation.

$5 x = - {(1 - y)}^{2} - 10$

Step 1.1.2

Divide each term in $5 x = - {(1 - y)}^{2} - 10$ by $5$ and simplify.

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Step 1.1.2.1

Divide each term in $5 x = - {(1 - y)}^{2} - 10$ by $5$ .

$\frac{5 x}{5} = \frac{- {(1 - y)}^{2}}{5} + \frac{- 10}{5}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $5$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- {(1 - y)}^{2}}{5} + \frac{- 10}{5}$

Step 1.1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- {(1 - y)}^{2}}{5} + \frac{- 10}{5}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

Move the negative in front of the fraction.

$x = - \frac{{(1 - y)}^{2}}{5} + \frac{- 10}{5}$

Step 1.1.2.3.1.2

Divide $- 10$ by $5$ .

$x = - \frac{{(1 - y)}^{2}}{5} - 2$

Step 1.1.3

Reorder terms.

$x = - \frac{1}{5} \cdot {(1 - y)}^{2} - 2$

Step 1.2

Complete the square for $- \frac{1}{5} \cdot {(1 - y)}^{2} - 2$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(1 - y)}^{2}$ as $(1 - y) (1 - y)$ .

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

Step 1.2.1.1.2

Expand $(1 - y) (1 - y)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

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

Step 1.2.1.1.2.2

Apply the distributive property.

$- \frac{1}{5} \cdot (1 \cdot 1 + 1 (- y) - y (1 - y)) - 2$

Step 1.2.1.1.2.3

Apply the distributive property.

$- \frac{1}{5} \cdot (1 \cdot 1 + 1 (- y) - y \cdot 1 - y (- y)) - 2$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $1$ by $1$ .

$- \frac{1}{5} \cdot (1 + 1 (- y) - y \cdot 1 - y (- y)) - 2$

Step 1.2.1.1.3.1.2

Multiply $- y$ by $1$ .

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

Step 1.2.1.1.3.1.3

Multiply $- 1$ by $1$ .

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

Step 1.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

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

Step 1.2.1.1.3.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 1.2.1.1.3.1.5.1

Move $y$ .

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

Step 1.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

$- \frac{1}{5} \cdot (1 - y - y - 1 \cdot - 1 y^{2}) - 2$

Step 1.2.1.1.3.1.6

Multiply $- 1$ by $- 1$ .

$- \frac{1}{5} \cdot (1 - y - y + 1 y^{2}) - 2$

Step 1.2.1.1.3.1.7

Multiply $y^{2}$ by $1$ .

$- \frac{1}{5} \cdot (1 - y - y + y^{2}) - 2$

Step 1.2.1.1.3.2

Subtract $y$ from $- y$ .

$- \frac{1}{5} \cdot (1 - 2 y + y^{2}) - 2$

Step 1.2.1.1.4

Apply the distributive property.

$- \frac{1}{5} \cdot 1 - \frac{1}{5} (- 2 y) - \frac{1}{5} y^{2} - 2$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Multiply $- 1$ by $1$ .

$- \frac{1}{5} - \frac{1}{5} (- 2 y) - \frac{1}{5} y^{2} - 2$

Step 1.2.1.1.5.2

Multiply $- \frac{1}{5} (- 2 y)$ .

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Step 1.2.1.1.5.2.1

Multiply $- 2$ by $- 1$ .

$- \frac{1}{5} + 2 (\frac{1}{5}) y - \frac{1}{5} y^{2} - 2$

Step 1.2.1.1.5.2.2

Combine $2$ and $\frac{1}{5}$ .

$- \frac{1}{5} + \frac{2}{5} y - \frac{1}{5} y^{2} - 2$

Step 1.2.1.1.5.2.3

Combine $\frac{2}{5}$ and $y$ .

$- \frac{1}{5} + \frac{2 y}{5} - \frac{1}{5} y^{2} - 2$

Step 1.2.1.1.5.3

Combine $y^{2}$ and $\frac{1}{5}$ .

$- \frac{1}{5} + \frac{2 y}{5} - \frac{y^{2}}{5} - 2$

Step 1.2.1.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{2 y}{5} - \frac{y^{2}}{5} - \frac{1}{5} - 2 \cdot \frac{5}{5}$

Step 1.2.1.3

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

$\frac{2 y}{5} - \frac{y^{2}}{5} - \frac{1}{5} + \frac{- 2 \cdot 5}{5}$

Step 1.2.1.4

Combine the numerators over the common denominator.

$\frac{2 y}{5} - \frac{y^{2}}{5} + \frac{- 1 - 2 \cdot 5}{5}$

Step 1.2.1.5

Simplify the numerator.

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Step 1.2.1.5.1

Multiply $- 2$ by $5$ .

$\frac{2 y}{5} - \frac{y^{2}}{5} + \frac{- 1 - 10}{5}$

Step 1.2.1.5.2

Subtract $10$ from $- 1$ .

$\frac{2 y}{5} - \frac{y^{2}}{5} + \frac{- 11}{5}$

Step 1.2.1.6

Move the negative in front of the fraction.

$\frac{2 y}{5} - \frac{y^{2}}{5} - \frac{11}{5}$

Step 1.2.1.7

Reorder $\frac{2 y}{5}$ and $- \frac{y^{2}}{5}$ .

$- \frac{y^{2}}{5} + \frac{2 y}{5} - \frac{11}{5}$

Step 1.2.2

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

$a = - \frac{1}{5}$

$b = \frac{2}{5}$

$c = - \frac{11}{5}$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

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Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.4.2.2

Cancel the common factor of $2$ .

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Step 1.2.4.2.2.1

Cancel the common factor.

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

Step 1.2.4.2.2.2

Rewrite the expression.

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

Step 1.2.4.2.3

Multiply $\frac{1}{5}$ by $\frac{1}{- \frac{1}{5}}$ .

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

Step 1.2.4.2.4

Multiply $- 1$ by $5$ .

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

Step 1.2.4.2.5

Combine $- 5$ and $\frac{1}{5}$ .

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

Step 1.2.4.2.6

Cancel the common factor of $- 5$ and $5$ .

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Step 1.2.4.2.6.1

Factor $5$ out of $- 5$ .

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

Step 1.2.4.2.6.2

Cancel the common factors.

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Step 1.2.4.2.6.2.1

Factor $5$ out of $5$ .

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

Step 1.2.4.2.6.2.2

Cancel the common factor.

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

Step 1.2.4.2.6.2.3

Rewrite the expression.

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

Step 1.2.4.2.6.2.4

Divide $- 1$ by $1$ .

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

Step 1.2.4.2.7

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$d = - 1 \cdot 1$

Step 1.2.4.2.8

Multiply $- 1$ by $1$ .

$d = - 1$

Step 1.2.5

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

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Step 1.2.5.1

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

$e = - \frac{11}{5} - \frac{{(\frac{2}{5})}^{2}}{4 (- \frac{1}{5})}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Simplify the numerator.

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Step 1.2.5.2.1.1.1

Apply the product rule to $\frac{2}{5}$ .

$e = - \frac{11}{5} - \frac{\frac{2^{2}}{5^{2}}}{4 (- \frac{1}{5})}$

Step 1.2.5.2.1.1.2

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

$e = - \frac{11}{5} - \frac{\frac{4}{5^{2}}}{4 (- \frac{1}{5})}$

Step 1.2.5.2.1.1.3

Raise $5$ to the power of $2$ .

$e = - \frac{11}{5} - \frac{\frac{4}{25}}{4 (- \frac{1}{5})}$

Step 1.2.5.2.1.2

Simplify the denominator.

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Step 1.2.5.2.1.2.1

Multiply $- 1$ by $4$ .

$e = - \frac{11}{5} - \frac{\frac{4}{25}}{- 4 (\frac{1}{5})}$

Step 1.2.5.2.1.2.2

Combine $- 4$ and $\frac{1}{5}$ .

$e = - \frac{11}{5} - \frac{\frac{4}{25}}{\frac{- 4}{5}}$

Step 1.2.5.2.1.3

Move the negative in front of the fraction.

$e = - \frac{11}{5} - \frac{\frac{4}{25}}{- \frac{4}{5}}$

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = - \frac{11}{5} - (\frac{4}{25} (- \frac{5}{4}))$

Step 1.2.5.2.1.5

Cancel the common factor of $4$ .

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Step 1.2.5.2.1.5.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$e = - \frac{11}{5} - (\frac{4}{25} \cdot \frac{- 5}{4})$

Step 1.2.5.2.1.5.2

Cancel the common factor.

$e = - \frac{11}{5} - (\frac{4}{25} \cdot \frac{- 5}{4})$

Step 1.2.5.2.1.5.3

Rewrite the expression.

$e = - \frac{11}{5} - (\frac{1}{25} \cdot - 5)$

Step 1.2.5.2.1.6

Cancel the common factor of $5$ .

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Step 1.2.5.2.1.6.1

Factor $5$ out of $25$ .

$e = - \frac{11}{5} - (\frac{1}{5 (5)} \cdot - 5)$

Step 1.2.5.2.1.6.2

Factor $5$ out of $- 5$ .

$e = - \frac{11}{5} - (\frac{1}{5 \cdot 5} \cdot (5 \cdot - 1))$

Step 1.2.5.2.1.6.3

Cancel the common factor.

$e = - \frac{11}{5} - (\frac{1}{5 \cdot 5} \cdot (5 \cdot - 1))$

Step 1.2.5.2.1.6.4

Rewrite the expression.

$e = - \frac{11}{5} - (\frac{1}{5} \cdot - 1)$

Step 1.2.5.2.1.7

Combine $\frac{1}{5}$ and $- 1$ .

$e = - \frac{11}{5} - \frac{- 1}{5}$

Step 1.2.5.2.1.8

Move the negative in front of the fraction.

$e = - \frac{11}{5} - - \frac{1}{5}$

Step 1.2.5.2.1.9

Multiply $- - \frac{1}{5}$ .

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Step 1.2.5.2.1.9.1

Multiply $- 1$ by $- 1$ .

$e = - \frac{11}{5} + 1 (\frac{1}{5})$

Step 1.2.5.2.1.9.2

Multiply $\frac{1}{5}$ by $1$ .

$e = - \frac{11}{5} + \frac{1}{5}$

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{- 11 + 1}{5}$

Step 1.2.5.2.3

Add $- 11$ and $1$ .

$e = \frac{- 10}{5}$

Step 1.2.5.2.4

Divide $- 10$ by $5$ .

$e = - 2$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{5} {(y - 1)}^{2} - 2$ .

$- \frac{1}{5} {(y - 1)}^{2} - 2$

Step 1.3

Set $x$ equal to the new right side.

$x = - \frac{1}{5} \cdot {(y - 1)}^{2} - 2$

Step 2

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

$a = - \frac{1}{5}$

$h = - 2$

$k = 1$

Step 3

Find the vertex $(h, k)$ .

$(- 2, 1)$

Step 4