Examples

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

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

Step 1

Complete the square for

5 x^{2} + 2 x - 5

$5 x^{2} + 2 x - 5$ .

Tap for more steps...

Step 1.1

Use the form

a x^{2} + b x + c

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

a

$a$ ,

b

$b$ , and

c

$c$ .

a = 5

$a = 5$

b = 2

$b = 2$

c = - 5

$c = - 5$

Step 1.2

Consider the vertex form of a parabola.

a {(x + d)}^{2} + e

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

Step 1.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

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

Tap for more steps...

Step 1.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

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

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

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

Step 1.3.2

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 1.3.2.1

Cancel the common factor.

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

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

Step 1.3.2.2

Rewrite the expression.

d = \frac{1}{5}

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

d = \frac{1}{5}

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

d = \frac{1}{5}

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

Step 1.4

Find the value of

e

$e$ using the formula

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

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

Tap for more steps...

Step 1.4.1

Substitute the values of

c

$c$ ,

b

$b$ and

a

$a$ into the formula

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

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

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

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

Step 1.4.2

Simplify the right side.

Tap for more steps...

Step 1.4.2.1

Simplify each term.

Tap for more steps...

Step 1.4.2.1.1

Raise

2

$2$ to the power of

2

$2$ .

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

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

Step 1.4.2.1.2

Multiply

4

$4$ by

5

$5$ .

e = - 5 - \frac{4}{20}

$e = - 5 - \frac{4}{20}$

Step 1.4.2.1.3

Cancel the common factor of

4

$4$ and

20

$20$ .

Tap for more steps...

Step 1.4.2.1.3.1

Factor

4

$4$ out of

4

$4$ .

e = - 5 - \frac{4 (1)}{20}

$e = - 5 - \frac{4 (1)}{20}$

Step 1.4.2.1.3.2

Cancel the common factors.

Tap for more steps...

Step 1.4.2.1.3.2.1

Factor

4

$4$ out of

20

$20$ .

e = - 5 - \frac{4 \cdot 1}{4 \cdot 5}

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

Step 1.4.2.1.3.2.2

Cancel the common factor.

e = - 5 - \frac{4 \cdot 1}{4 \cdot 5}

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

Step 1.4.2.1.3.2.3

Rewrite the expression.

e = - 5 - \frac{1}{5}

$e = - 5 - \frac{1}{5}$

e = - 5 - \frac{1}{5}

$e = - 5 - \frac{1}{5}$

e = - 5 - \frac{1}{5}

$e = - 5 - \frac{1}{5}$

e = - 5 - \frac{1}{5}

$e = - 5 - \frac{1}{5}$

Step 1.4.2.2

To write

- 5

$- 5$ as a fraction with a common denominator, multiply by

\frac{5}{5}

$\frac{5}{5}$ .

e = - 5 \cdot \frac{5}{5} - \frac{1}{5}

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

Step 1.4.2.3

Combine

- 5

$- 5$ and

\frac{5}{5}

$\frac{5}{5}$ .

e = \frac{- 5 \cdot 5}{5} - \frac{1}{5}

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

Step 1.4.2.4

Combine the numerators over the common denominator.

e = \frac{- 5 \cdot 5 - 1}{5}

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

Step 1.4.2.5

Simplify the numerator.

Tap for more steps...

Step 1.4.2.5.1

Multiply

- 5

$- 5$ by

5

$5$ .

e = \frac{- 25 - 1}{5}

$e = \frac{- 25 - 1}{5}$

Step 1.4.2.5.2

Subtract

1

$1$ from

- 25

$- 25$ .

e = \frac{- 26}{5}

$e = \frac{- 26}{5}$

e = \frac{- 26}{5}

$e = \frac{- 26}{5}$

Step 1.4.2.6

Move the negative in front of the fraction.

e = - \frac{26}{5}

$e = - \frac{26}{5}$

e = - \frac{26}{5}

$e = - \frac{26}{5}$

e = - \frac{26}{5}

$e = - \frac{26}{5}$

Step 1.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

5 {(x + \frac{1}{5})}^{2} - \frac{26}{5}

$5 {(x + \frac{1}{5})}^{2} - \frac{26}{5}$ .

5 {(x + \frac{1}{5})}^{2} - \frac{26}{5}

$5 {(x + \frac{1}{5})}^{2} - \frac{26}{5}$

5 {(x + \frac{1}{5})}^{2} - \frac{26}{5}

$5 {(x + \frac{1}{5})}^{2} - \frac{26}{5}$

Step 2

Set

y

$y$ equal to the new right side.

y = 5 {(x + \frac{1}{5})}^{2} - \frac{26}{5}

$y = 5 {(x + \frac{1}{5})}^{2} - \frac{26}{5}$

Enter YOUR Problem