Find the Standard Form of the Parabola y=x^2-7x+18

$y = x^{2} - 7 x + 18$

Step 1

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

$x^{2} - 7 x + 18 = y$

Step 2

Subtract $18$ from both sides of the equation.

$x^{2} - 7 x = y - 18$

Step 3

Complete the square.

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

$a = 1$

$b = - 7$

$c = 0 = y - 18$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 18$

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

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

$d = \frac{- 7}{2 \cdot 1}$

Simplify the right side.

Multiply $2$ by $1$ .

$d = \frac{- 7}{2}$

Move the negative in front of the fraction.

$d = - \frac{7}{2}$

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 $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 7)}^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $- 7$ to the power of $2$ .

$e = 0 - \frac{49}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{49}{4}$

Subtract $\frac{49}{4}$ from $0$ .

$e = - \frac{49}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - \frac{7}{2})}^{2} - \frac{49}{4}$ .

${(x - \frac{7}{2})}^{2} - \frac{49}{4} = y - 18$

Step 4

Move all terms not containing $x$ to the right side of the equation.

Add $\frac{49}{4}$ to both sides of the equation.

${(x - \frac{7}{2})}^{2} = y - 18 + \frac{49}{4}$

To write $- 18$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

${(x - \frac{7}{2})}^{2} = y - 18 \cdot \frac{4}{4} + \frac{49}{4}$

Combine $- 18$ and $\frac{4}{4}$ .

${(x - \frac{7}{2})}^{2} = y + \frac{- 18 \cdot 4}{4} + \frac{49}{4}$

Combine the numerators over the common denominator.

${(x - \frac{7}{2})}^{2} = y + \frac{- 18 \cdot 4 + 49}{4}$

Simplify the numerator.

Multiply $- 18$ by $4$ .

${(x - \frac{7}{2})}^{2} = y + \frac{- 72 + 49}{4}$

Add $- 72$ and $49$ .

${(x - \frac{7}{2})}^{2} = y + \frac{- 23}{4}$

Move the negative in front of the fraction.

${(x - \frac{7}{2})}^{2} = y - \frac{23}{4}$

Step 5

Factor.

To write $y$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

${(x - \frac{7}{2})}^{2} = y \cdot \frac{4}{4} - \frac{23}{4}$

Combine $y$ and $\frac{4}{4}$ .

${(x - \frac{7}{2})}^{2} = \frac{y \cdot 4}{4} - \frac{23}{4}$

Combine the numerators over the common denominator.

${(x - \frac{7}{2})}^{2} = \frac{y \cdot 4 - 23}{4}$

Move $4$ to the left of $y$ .

${(x - \frac{7}{2})}^{2} = \frac{4 y - 23}{4}$

Reorder terms.

${(x - \frac{7}{2})}^{2} = \frac{1}{4} (4 y - 23)$

Find the Standard Form of the Parabola y=x^2-7x+18

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