Find the Standard Form of the Parabola f(x)=5x^2-5x+1

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

Step 1

Write $f (x) = 5 x^{2} - 5 x + 1$ as an equation.

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

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.

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

Step 3

Subtract $1$ from both sides of the equation.

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

Step 4

Complete the square.

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

$a = 5$

$b = - 5$

$c = 0 = y - 1$

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y - 1$

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

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

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

Simplify the right side.

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

Factor $5$ out of $- 5$ .

$d = \frac{5 \cdot - 1}{2 \cdot 5}$

Cancel the common factors.

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

$d = \frac{5 \cdot - 1}{5 \cdot 2}$

Cancel the common factor.

$d = \frac{5 \cdot - 1}{5 \cdot 2}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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 $5$ .

Rewrite $- 5$ as $- 1 (5)$ .

$e = 0 - \frac{{(- 1 (5))}^{2}}{4 \cdot 5}$

Apply the product rule to $- 1 (5)$ .

$e = 0 - \frac{{(- 1)}^{2} \cdot 5^{2}}{4 \cdot 5}$

Raise $- 1$ to the power of $2$ .

$e = 0 - \frac{1 \cdot 5^{2}}{4 \cdot 5}$

Multiply $5^{2}$ by $1$ .

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

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 $5 {(x - \frac{1}{2})}^{2} - \frac{5}{4}$ .

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

Step 5

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

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

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

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

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

Combine $- 1$ and $\frac{4}{4}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $4$ .

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

Add $- 4$ and $5$ .

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

Step 6

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

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

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

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Simplify each term.

Multiply the numerator by the reciprocal of the denominator.

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

Multiply $\frac{1}{4} \cdot \frac{1}{5}$ .

Multiply $\frac{1}{4}$ by $\frac{1}{5}$ .

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

Multiply $4$ by $5$ .

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

Step 7

Factor.

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

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

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

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

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

Multiply $5$ by $4$ .

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

Combine the numerators over the common denominator.

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

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

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

Reorder terms.

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

Find the Standard Form of the Parabola f(x)=5x^2-5x+1

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