Find the Standard Form of the Parabola y=-5x^2-30x+35

$y = - 5 x^{2} - 30 x + 35$

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

$- 5 x^{2} - 30 x + 35 = y$

Subtract $35$ from both sides of the equation.

$- 5 x^{2} - 30 x = y - 35$

Complete the square.

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

$a = - 5$

$b = - 30$

$c = 0 = y - 35$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 35$

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

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

$d = \frac{- 30}{2 \cdot - 5}$

Simplify the right side.

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

Factor $2$ out of $- 30$ .

$d = \frac{2 \cdot - 15}{2 \cdot - 5}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 5$ .

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

Cancel the common factor.

$d = \frac{2 \cdot - 15}{2 \cdot - 5}$

Rewrite the expression.

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

Cancel the common factor of $- 15$ and $- 5$ .

Factor $- 5$ out of $- 15$ .

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

Cancel the common factors.

Factor $- 5$ out of $- 5$ .

$d = \frac{- 5 \cdot 3}{- 5 \cdot 1}$

Cancel the common factor.

$d = \frac{- 5 \cdot 3}{- 5 \cdot 1}$

Rewrite the expression.

$d = \frac{3}{1}$

Divide $3$ by $1$ .

$d = 3$

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

$e = 0 - \frac{{(- 30)}^{2}}{4 \cdot - 5}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{900}{4 \cdot - 5}$

Multiply $4$ by $- 5$ .

$e = 0 - \frac{900}{- 20}$

Divide $900$ by $- 20$ .

$e = 0 - - 45$

Multiply $- 1$ by $- 45$ .

$e = 0 + 45$

Add $0$ and $45$ .

$e = 45$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 5 {(x + 3)}^{2} + 45$ .

$- 5 {(x + 3)}^{2} + 45 = y - 35$

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

Subtract $45$ from both sides of the equation.

$- 5 {(x + 3)}^{2} = y - 35 - 45$

Subtract $45$ from $- 35$ .

$- 5 {(x + 3)}^{2} = y - 80$

Divide each term in $- 5 {(x + 3)}^{2} = y - 80$ by $- 5$ and simplify.

Divide each term in $- 5 {(x + 3)}^{2} = y - 80$ by $- 5$ .

$\frac{- 5 {(x + 3)}^{2}}{- 5} = \frac{y}{- 5} + \frac{- 80}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 {(x + 3)}^{2}}{- 5} = \frac{y}{- 5} + \frac{- 80}{- 5}$

Divide ${(x + 3)}^{2}$ by $1$ .

${(x + 3)}^{2} = \frac{y}{- 5} + \frac{- 80}{- 5}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

${(x + 3)}^{2} = - \frac{y}{5} + \frac{- 80}{- 5}$

Divide $- 80$ by $- 5$ .

${(x + 3)}^{2} = - \frac{y}{5} + 16$

Factor.

To write $16$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

${(x + 3)}^{2} = - \frac{y}{5} + 16 \cdot \frac{5}{5}$

Combine $16$ and $\frac{5}{5}$ .

${(x + 3)}^{2} = - \frac{y}{5} + \frac{16 \cdot 5}{5}$

Combine the numerators over the common denominator.

${(x + 3)}^{2} = \frac{- y + 16 \cdot 5}{5}$

Multiply $16$ by $5$ .

${(x + 3)}^{2} = \frac{- y + 80}{5}$

Factor $- 1$ out of $- y$ .

${(x + 3)}^{2} = \frac{- (y) + 80}{5}$

Rewrite $80$ as $- 1 (- 80)$ .

${(x + 3)}^{2} = \frac{- (y) - 1 (- 80)}{5}$

Factor $- 1$ out of $- (y) - 1 (- 80)$ .

${(x + 3)}^{2} = \frac{- (y - 80)}{5}$

Move the negative in front of the fraction.

${(x + 3)}^{2} = - \frac{y - 80}{5}$

Reorder terms.

${(x + 3)}^{2} = - (\frac{1}{5} (y - 80))$

Remove parentheses.

${(x + 3)}^{2} = - \frac{1}{5} (y - 80)$

Find the Standard Form of the Parabola y=-5x^2-30x+35

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