Precalculus Examples

$x^{2} + 10 x + 11$

Step 1

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

$a = 1$

$b = 10$

$c = 11$

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

$d = \frac{10}{2 \cdot 1}$

Step 3.2

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

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Step 3.2.1

Factor $2$ out of $10$ .

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

Step 3.2.2

Cancel the common factors.

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Step 3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot 5}{2 (1)}$

Step 3.2.2.2

Cancel the common factor.

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

Step 3.2.2.3

Rewrite the expression.

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

Step 3.2.2.4

Divide $5$ by $1$ .

$d = 5$

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

$e = 11 - \frac{10^{2}}{4 \cdot 1}$

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 $10$ to the power of $2$ .

$e = 11 - \frac{100}{4 \cdot 1}$

Step 4.2.1.2

Multiply $4$ by $1$ .

$e = 11 - \frac{100}{4}$

Step 4.2.1.3

Divide $100$ by $4$ .

$e = 11 - 1 \cdot 25$

Step 4.2.1.4

Multiply $- 1$ by $25$ .

$e = 11 - 25$

Step 4.2.2

Subtract $25$ from $11$ .

$e = - 14$

Step 5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + 5)}^{2} - 14$ .

${(x + 5)}^{2} - 14$