Algebra Examples

$x^{2} - 5 x + \frac{49}{25}$

Step 1

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

$a = 1$

$b = - 5$

$c = \frac{49}{25}$

Step 2

Consider the vertex form of a parabola.

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

Step 3

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

Tap for more steps...

Step 3.1

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

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

Step 3.2

Simplify the right side.

Tap for more steps...

Step 3.2.1

Multiply $2$ by $1$ .

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

Step 3.2.2

Move the negative in front of the fraction.

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

Step 4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

Tap for more steps...

Step 4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = \frac{49}{25} - \frac{{(- 5)}^{2}}{4 \cdot 1}$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Raise $- 5$ to the power of $2$ .

$e = \frac{49}{25} - \frac{25}{4 \cdot 1}$

Step 4.2.1.2

Multiply $4$ by $1$ .

$e = \frac{49}{25} - \frac{25}{4}$

Step 4.2.2

To write $\frac{49}{25}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = \frac{49}{25} \cdot \frac{4}{4} - \frac{25}{4}$

Step 4.2.3

To write $- \frac{25}{4}$ as a fraction with a common denominator, multiply by $\frac{25}{25}$ .

$e = \frac{49}{25} \cdot \frac{4}{4} - \frac{25}{4} \cdot \frac{25}{25}$

Step 4.2.4

Write each expression with a common denominator of $100$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.2.4.1

Multiply $\frac{49}{25}$ by $\frac{4}{4}$ .

$e = \frac{49 \cdot 4}{25 \cdot 4} - \frac{25}{4} \cdot \frac{25}{25}$

Step 4.2.4.2

Multiply $25$ by $4$ .

$e = \frac{49 \cdot 4}{100} - \frac{25}{4} \cdot \frac{25}{25}$

Step 4.2.4.3

Multiply $\frac{25}{4}$ by $\frac{25}{25}$ .

$e = \frac{49 \cdot 4}{100} - \frac{25 \cdot 25}{4 \cdot 25}$

Step 4.2.4.4

Multiply $4$ by $25$ .

$e = \frac{49 \cdot 4}{100} - \frac{25 \cdot 25}{100}$

Step 4.2.5

Combine the numerators over the common denominator.

$e = \frac{49 \cdot 4 - 25 \cdot 25}{100}$

Step 4.2.6

Simplify the numerator.

Tap for more steps...

Step 4.2.6.1

Multiply $49$ by $4$ .

$e = \frac{196 - 25 \cdot 25}{100}$

Step 4.2.6.2

Multiply $- 25$ by $25$ .

$e = \frac{196 - 625}{100}$

Step 4.2.6.3

Subtract $625$ from $196$ .

$e = \frac{- 429}{100}$

Step 4.2.7

Move the negative in front of the fraction.

$e = - \frac{429}{100}$

Step 5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - \frac{5}{2})}^{2} - \frac{429}{100}$ .

${(x - \frac{5}{2})}^{2} - \frac{429}{100}$