Find the Standard Form of the Parabola y=5x^2+5x-10

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

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.

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

Step 2

Add $10$ to both sides of the equation.

$5 x^{2} + 5 x = y + 10$

Step 3

Complete the square.

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

$a = 5$

$b = 5$

$c = 0 = y + 10$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y + 10$

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

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

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

Cancel the common factor of $5$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Simplify the right side.

Cancel the common factor of $5^{2}$ and $5$ .

Factor $5$ out of $5^{2}$ .

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

Cancel the common factors.

Factor $5$ out of $4 \cdot 5$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Subtract $\frac{5}{4}$ from $0$ .

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

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $5 {(x + \frac{1}{2})}^{2} - \frac{5}{4}$ .

$5 {(x + \frac{1}{2})}^{2} - \frac{5}{4} = y + 10$

Step 4

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

Add $\frac{5}{4}$ to both sides of the equation.

$5 {(x + \frac{1}{2})}^{2} = y + 10 + \frac{5}{4}$

To write $10$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$5 {(x + \frac{1}{2})}^{2} = y + 10 \cdot \frac{4}{4} + \frac{5}{4}$

Combine $10$ and $\frac{4}{4}$ .

$5 {(x + \frac{1}{2})}^{2} = y + \frac{10 \cdot 4}{4} + \frac{5}{4}$

Combine the numerators over the common denominator.

$5 {(x + \frac{1}{2})}^{2} = y + \frac{10 \cdot 4 + 5}{4}$

Simplify the numerator.

Multiply $10$ by $4$ .

$5 {(x + \frac{1}{2})}^{2} = y + \frac{40 + 5}{4}$

Add $40$ and $5$ .

$5 {(x + \frac{1}{2})}^{2} = y + \frac{45}{4}$

Step 5

Divide each term in $5 {(x + \frac{1}{2})}^{2} = y + \frac{45}{4}$ by $5$ and simplify.

Divide each term in $5 {(x + \frac{1}{2})}^{2} = y + \frac{45}{4}$ by $5$ .

$\frac{5 {(x + \frac{1}{2})}^{2}}{5} = \frac{y}{5} + \frac{\frac{45}{4}}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 {(x + \frac{1}{2})}^{2}}{5} = \frac{y}{5} + \frac{\frac{45}{4}}{5}$

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

${(x + \frac{1}{2})}^{2} = \frac{y}{5} + \frac{\frac{45}{4}}{5}$

Simplify the right side.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

${(x + \frac{1}{2})}^{2} = \frac{y}{5} + \frac{45}{4} \cdot \frac{1}{5}$

Cancel the common factor of $5$ .

Factor $5$ out of $45$ .

${(x + \frac{1}{2})}^{2} = \frac{y}{5} + \frac{5 (9)}{4} \cdot \frac{1}{5}$

Cancel the common factor.

${(x + \frac{1}{2})}^{2} = \frac{y}{5} + \frac{5 \cdot 9}{4} \cdot \frac{1}{5}$

Rewrite the expression.

${(x + \frac{1}{2})}^{2} = \frac{y}{5} + \frac{9}{4}$

Step 6

Factor.

To write $\frac{y}{5}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

${(x + \frac{1}{2})}^{2} = \frac{y}{5} \cdot \frac{4}{4} + \frac{9}{4}$

To write $\frac{9}{4}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

${(x + \frac{1}{2})}^{2} = \frac{y}{5} \cdot \frac{4}{4} + \frac{9}{4} \cdot \frac{5}{5}$

Write each expression with a common denominator of $20$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{y}{5}$ by $\frac{4}{4}$ .

${(x + \frac{1}{2})}^{2} = \frac{y \cdot 4}{5 \cdot 4} + \frac{9}{4} \cdot \frac{5}{5}$

Multiply $5$ by $4$ .

${(x + \frac{1}{2})}^{2} = \frac{y \cdot 4}{20} + \frac{9}{4} \cdot \frac{5}{5}$

Multiply $\frac{9}{4}$ by $\frac{5}{5}$ .

${(x + \frac{1}{2})}^{2} = \frac{y \cdot 4}{20} + \frac{9 \cdot 5}{4 \cdot 5}$

Multiply $4$ by $5$ .

${(x + \frac{1}{2})}^{2} = \frac{y \cdot 4}{20} + \frac{9 \cdot 5}{20}$

Combine the numerators over the common denominator.

${(x + \frac{1}{2})}^{2} = \frac{y \cdot 4 + 9 \cdot 5}{20}$

Simplify the numerator.

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

${(x + \frac{1}{2})}^{2} = \frac{4 y + 9 \cdot 5}{20}$

Multiply $9$ by $5$ .

${(x + \frac{1}{2})}^{2} = \frac{4 y + 45}{20}$

Reorder terms.

${(x + \frac{1}{2})}^{2} = \frac{1}{20} (4 y + 45)$

Find the Standard Form of the Parabola y=5x^2+5x-10

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