Find the Standard Form of the Parabola y=x^2+3x-10

$y = x^{2} + 3 x - 10$

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

$x^{2} + 3 x - 10 = y$

Add $10$ to both sides of the equation.

$x^{2} + 3 x = y + 10$

Complete the square.

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

$a = 1$

$b = 3$

$c = 0 = y + 10$

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 $d = \frac{b}{2 a}$ .

$d = \frac{3}{2 \cdot 1}$

Multiply $2$ by $1$ .

$d = \frac{3}{2}$

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

$e = 0 - \frac{3^{2}}{4 \cdot 1}$

Simplify the right side.

Simplify each term.

Raise $3$ to the power of $2$ .

$e = 0 - \frac{9}{4 \cdot 1}$

Multiply $4$ by $1$ .

$e = 0 - \frac{9}{4}$

Subtract $\frac{9}{4}$ from $0$ .

$e = - \frac{9}{4}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{3}{2})}^{2} - \frac{9}{4}$ .

${(x + \frac{3}{2})}^{2} - \frac{9}{4} = y + 10$

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

Add $\frac{9}{4}$ to both sides of the equation.

${(x + \frac{3}{2})}^{2} = y + 10 + \frac{9}{4}$

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

${(x + \frac{3}{2})}^{2} = y + 10 \cdot \frac{4}{4} + \frac{9}{4}$

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

${(x + \frac{3}{2})}^{2} = y + \frac{10 \cdot 4}{4} + \frac{9}{4}$

Combine the numerators over the common denominator.

${(x + \frac{3}{2})}^{2} = y + \frac{10 \cdot 4 + 9}{4}$

Simplify the numerator.

Multiply $10$ by $4$ .

${(x + \frac{3}{2})}^{2} = y + \frac{40 + 9}{4}$

Add $40$ and $9$ .

${(x + \frac{3}{2})}^{2} = y + \frac{49}{4}$

Factor.

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

${(x + \frac{3}{2})}^{2} = y \cdot \frac{4}{4} + \frac{49}{4}$

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

${(x + \frac{3}{2})}^{2} = \frac{y \cdot 4}{4} + \frac{49}{4}$

Combine the numerators over the common denominator.

${(x + \frac{3}{2})}^{2} = \frac{y \cdot 4 + 49}{4}$

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

${(x + \frac{3}{2})}^{2} = \frac{4 y + 49}{4}$

Reorder terms.

${(x + \frac{3}{2})}^{2} = \frac{1}{4} (4 y + 49)$

Find the Standard Form of the Parabola y=x^2+3x-10

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