Algebra Examples

$\frac{1}{4} x^{2} + x + 1$

Step 1

Combine $\frac{1}{4}$ and $x^{2}$ .

$\frac{x^{2}}{4} + x + 1$

Step 2

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

$a = \frac{1}{4}$

$b = 1$

$c = 1$

Step 3

Consider the vertex form of a parabola.

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

Step 4

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

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Step 4.1

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

$d = \frac{1}{2 (\frac{1}{4})}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Combine $2$ and $\frac{1}{4}$ .

$d = \frac{1}{\frac{2}{4}}$

Step 4.2.2

Cancel the common factor of $2$ and $4$ .

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Step 4.2.2.1

Factor $2$ out of $2$ .

$d = \frac{1}{\frac{2 (1)}{4}}$

Step 4.2.2.2

Cancel the common factors.

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Step 4.2.2.2.1

Factor $2$ out of $4$ .

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

Step 4.2.2.2.2

Cancel the common factor.

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

Step 4.2.2.2.3

Rewrite the expression.

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

Step 4.2.3

Multiply the numerator by the reciprocal of the denominator.

$d = 1 \cdot 2$

Step 4.2.4

Multiply $2$ by $1$ .

$d = 2$

Step 5

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

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Step 5.1

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

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

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

One to any power is one.

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

Step 5.2.1.2

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

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

Step 5.2.1.3

Divide $4$ by $4$ .

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

Step 5.2.1.4

Cancel the common factor of $1$ .

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Step 5.2.1.4.1

Cancel the common factor.

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

Step 5.2.1.4.2

Rewrite the expression.

$e = 1 - 1 \cdot 1$

Step 5.2.1.5

Multiply $- 1$ by $1$ .

$e = 1 - 1$

Step 5.2.2

Subtract $1$ from $1$ .

$e = 0$

Step 6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{4} {(x + 2)}^{2} + 0$ .

$\frac{1}{4} {(x + 2)}^{2} + 0$