Find the Standard Form of the Parabola f(x)=5x^2-10x

$f (x) = 5 x^{2} - 10 x$

Step 1

Write $f (x) = 5 x^{2} - 10 x$ as an equation.

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

Step 2

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

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

Step 3

Complete the square.

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

$a = 5$

$b = - 10$

$c = 0 = y$

Consider the vertex form of a parabola.

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

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

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

$d = \frac{- 10}{2 \cdot 5}$

Simplify the right side.

Cancel the common factor of $- 10$ and $2$ .

Factor $2$ out of $- 10$ .

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

Factor $5$ out of $- 5$ .

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

Cancel the common factors.

Factor $5$ out of $5$ .

$d = \frac{5 \cdot - 1}{5 (1)}$

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- 1$ by $1$ .

$d = - 1$

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

$e = 0 - \frac{{(- 10)}^{2}}{4 \cdot 5}$

Simplify the right side.

Simplify each term.

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

$e = 0 - \frac{100}{4 \cdot 5}$

Multiply $4$ by $5$ .

$e = 0 - \frac{100}{20}$

Divide $100$ by $20$ .

$e = 0 - 1 \cdot 5$

Multiply $- 1$ by $5$ .

$e = 0 - 5$

Subtract $5$ from $0$ .

$e = - 5$

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

$5 {(x - 1)}^{2} - 5 = y$

Step 4

Add $5$ to both sides of the equation.

$5 {(x - 1)}^{2} = y + 5$

Step 5

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

Divide each term in $5 {(x - 1)}^{2} = y + 5$ by $5$ .

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Divide $5$ by $5$ .

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

Step 6

Factor.

Write $1$ as a fraction with a common denominator.

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

Combine the numerators over the common denominator.

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

Reorder terms.

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

Find the Standard Form of the Parabola f(x)=5x^2-10x

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