Algebra Examples

$x^{2} - 9 x + 14$

Step 1

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

$a = 1$

$b = - 9$

$c = 14$

Step 2

Consider the vertex form of a parabola.

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

Step 3

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

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Step 3.1

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

$d = \frac{- 9}{2 \cdot 1}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $2$ by $1$ .

$d = \frac{- 9}{2}$

Step 3.2.2

Move the negative in front of the fraction.

$d = - \frac{9}{2}$

Step 4

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

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Step 4.1

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

$e = 14 - \frac{{(- 9)}^{2}}{4 \cdot 1}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raise $- 9$ to the power of $2$ .

$e = 14 - \frac{81}{4 \cdot 1}$

Step 4.2.1.2

Multiply $4$ by $1$ .

$e = 14 - \frac{81}{4}$

Step 4.2.2

To write $14$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 14 \cdot \frac{4}{4} - \frac{81}{4}$

Step 4.2.3

Combine $14$ and $\frac{4}{4}$ .

$e = \frac{14 \cdot 4}{4} - \frac{81}{4}$

Step 4.2.4

Combine the numerators over the common denominator.

$e = \frac{14 \cdot 4 - 81}{4}$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $14$ by $4$ .

$e = \frac{56 - 81}{4}$

Step 4.2.5.2

Subtract $81$ from $56$ .

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

Step 4.2.6

Move the negative in front of the fraction.

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

Step 5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - \frac{9}{2})}^{2} - \frac{25}{4}$ .

${(x - \frac{9}{2})}^{2} - \frac{25}{4}$