Find the Vertex Form (y-1)^2=20(x-3)

${(y - 1)}^{2} = 20 (x - 3)$

Step 1

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

Rewrite the equation as $20 (x - 3) = {(y - 1)}^{2}$ .

$20 (x - 3) = {(y - 1)}^{2}$

Divide each term in $20 (x - 3) = {(y - 1)}^{2}$ by $20$ and simplify.

Divide each term in $20 (x - 3) = {(y - 1)}^{2}$ by $20$ .

$\frac{20 (x - 3)}{20} = \frac{{(y - 1)}^{2}}{20}$

Simplify the left side.

Cancel the common factor of $20$ .

Cancel the common factor.

$\frac{20 (x - 3)}{20} = \frac{{(y - 1)}^{2}}{20}$

Divide $x - 3$ by $1$ .

$x - 3 = \frac{{(y - 1)}^{2}}{20}$

Add $3$ to both sides of the equation.

$x = \frac{{(y - 1)}^{2}}{20} + 3$

Reorder terms.

$x = \frac{1}{20} \cdot {(y - 1)}^{2} + 3$

Step 2

Complete the square for $\frac{1}{20} \cdot {(y - 1)}^{2} + 3$ .

Simplify the expression.

Simplify each term.

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

$\frac{1}{20} \cdot ((y - 1) (y - 1)) + 3$

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

Apply the distributive property.

$\frac{1}{20} \cdot (y (y - 1) - 1 (y - 1)) + 3$

Apply the distributive property.

$\frac{1}{20} \cdot (y \cdot y + y \cdot - 1 - 1 (y - 1)) + 3$

Apply the distributive property.

$\frac{1}{20} \cdot (y \cdot y + y \cdot - 1 - 1 y - 1 \cdot - 1) + 3$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$\frac{1}{20} \cdot (y^{2} + y \cdot - 1 - 1 y - 1 \cdot - 1) + 3$

Move $- 1$ to the left of $y$ .

$\frac{1}{20} \cdot (y^{2} - 1 \cdot y - 1 y - 1 \cdot - 1) + 3$

Rewrite $- 1 y$ as $- y$ .

$\frac{1}{20} \cdot (y^{2} - y - 1 y - 1 \cdot - 1) + 3$

Rewrite $- 1 y$ as $- y$ .

$\frac{1}{20} \cdot (y^{2} - y - y - 1 \cdot - 1) + 3$

Multiply $- 1$ by $- 1$ .

$\frac{1}{20} \cdot (y^{2} - y - y + 1) + 3$

Subtract $y$ from $- y$ .

$\frac{1}{20} \cdot (y^{2} - 2 y + 1) + 3$

Apply the distributive property.

$\frac{1}{20} y^{2} + \frac{1}{20} (- 2 y) + \frac{1}{20} \cdot 1 + 3$

Simplify.

Combine $\frac{1}{20}$ and $y^{2}$ .

$\frac{y^{2}}{20} + \frac{1}{20} (- 2 y) + \frac{1}{20} \cdot 1 + 3$

Cancel the common factor of $2$ .

Factor $2$ out of $20$ .

$\frac{y^{2}}{20} + \frac{1}{2 (10)} (- 2 y) + \frac{1}{20} \cdot 1 + 3$

Factor $2$ out of $- 2 y$ .

$\frac{y^{2}}{20} + \frac{1}{2 (10)} (2 (- y)) + \frac{1}{20} \cdot 1 + 3$

Cancel the common factor.

$\frac{y^{2}}{20} + \frac{1}{2 \cdot 10} (2 (- y)) + \frac{1}{20} \cdot 1 + 3$

Rewrite the expression.

$\frac{y^{2}}{20} + \frac{1}{10} (- y) + \frac{1}{20} \cdot 1 + 3$

Combine $\frac{1}{10}$ and $y$ .

$\frac{y^{2}}{20} - \frac{y}{10} + \frac{1}{20} \cdot 1 + 3$

Multiply $\frac{1}{20}$ by $1$ .

$\frac{y^{2}}{20} - \frac{y}{10} + \frac{1}{20} + 3$

To write $3$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$\frac{y^{2}}{20} - \frac{y}{10} + \frac{1}{20} + 3 \cdot \frac{20}{20}$

Combine $3$ and $\frac{20}{20}$ .

$\frac{y^{2}}{20} - \frac{y}{10} + \frac{1}{20} + \frac{3 \cdot 20}{20}$

Combine the numerators over the common denominator.

$\frac{y^{2}}{20} - \frac{y}{10} + \frac{1 + 3 \cdot 20}{20}$

Simplify the numerator.

Multiply $3$ by $20$ .

$\frac{y^{2}}{20} - \frac{y}{10} + \frac{1 + 60}{20}$

Add $1$ and $60$ .

$\frac{y^{2}}{20} - \frac{y}{10} + \frac{61}{20}$

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

$a = \frac{1}{20}$

$b = - \frac{1}{10}$

$c = \frac{61}{20}$

Consider the vertex form of a parabola.

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

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

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

$d = \frac{- \frac{1}{10}}{2 (\frac{1}{20})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{1}{10} \cdot \frac{1}{2 (\frac{1}{20})}$

Combine $2$ and $\frac{1}{20}$ .

$d = - \frac{1}{10} \cdot \frac{1}{\frac{2}{20}}$

Cancel the common factor of $2$ and $20$ .

Factor $2$ out of $2$ .

$d = - \frac{1}{10} \cdot \frac{1}{\frac{2 (1)}{20}}$

Cancel the common factors.

Factor $2$ out of $20$ .

$d = - \frac{1}{10} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 10}}$

Cancel the common factor.

$d = - \frac{1}{10} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 10}}$

Rewrite the expression.

$d = - \frac{1}{10} \cdot \frac{1}{\frac{1}{10}}$

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $10$ .

Move the leading negative in $- \frac{1}{10}$ into the numerator.

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

Factor $10$ out of $1 \cdot 10$ .

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

Cancel the common factor.

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

Rewrite the expression.

$d = - 1$

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

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

$e = \frac{61}{20} - \frac{{(- \frac{1}{10})}^{2}}{4 (\frac{1}{20})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $- \frac{1}{10}$ .

$e = \frac{61}{20} - \frac{{(- 1)}^{2} {(\frac{1}{10})}^{2}}{4 (\frac{1}{20})}$

Raise $- 1$ to the power of $2$ .

$e = \frac{61}{20} - \frac{1 {(\frac{1}{10})}^{2}}{4 (\frac{1}{20})}$

Apply the product rule to $\frac{1}{10}$ .

$e = \frac{61}{20} - \frac{1 \frac{1^{2}}{10^{2}}}{4 (\frac{1}{20})}$

One to any power is one.

$e = \frac{61}{20} - \frac{1 \frac{1}{10^{2}}}{4 (\frac{1}{20})}$

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

$e = \frac{61}{20} - \frac{1 (\frac{1}{100})}{4 (\frac{1}{20})}$

Multiply $\frac{1}{100}$ by $1$ .

$e = \frac{61}{20} - \frac{\frac{1}{100}}{4 (\frac{1}{20})}$

Combine $4$ and $\frac{1}{20}$ .

$e = \frac{61}{20} - \frac{\frac{1}{100}}{\frac{4}{20}}$

Cancel the common factor of $4$ and $20$ .

Factor $4$ out of $4$ .

$e = \frac{61}{20} - \frac{\frac{1}{100}}{\frac{4 (1)}{20}}$

Cancel the common factors.

Factor $4$ out of $20$ .

$e = \frac{61}{20} - \frac{\frac{1}{100}}{\frac{4 \cdot 1}{4 \cdot 5}}$

Cancel the common factor.

$e = \frac{61}{20} - \frac{\frac{1}{100}}{\frac{4 \cdot 1}{4 \cdot 5}}$

Rewrite the expression.

$e = \frac{61}{20} - \frac{\frac{1}{100}}{\frac{1}{5}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{61}{20} - (\frac{1}{100} \cdot 5)$

Cancel the common factor of $5$ .

Factor $5$ out of $100$ .

$e = \frac{61}{20} - (\frac{1}{5 (20)} \cdot 5)$

Cancel the common factor.

$e = \frac{61}{20} - (\frac{1}{5 \cdot 20} \cdot 5)$

Rewrite the expression.

$e = \frac{61}{20} - \frac{1}{20}$

Combine the numerators over the common denominator.

$e = \frac{61 - 1}{20}$

Subtract $1$ from $61$ .

$e = \frac{60}{20}$

Divide $60$ by $20$ .

$e = 3$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{20} {(y - 1)}^{2} + 3$ .

$\frac{1}{20} {(y - 1)}^{2} + 3$

Step 3

Set $x$ equal to the new right side.

$x = \frac{1}{20} \cdot {(y - 1)}^{2} + 3$

Find the Vertex Form (y-1)^2=20(x-3)

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