Algebra Examples

$3 + x - x^{2}$

Step 1

Simplify the expression.

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Step 1.1

Move $3$ .

$x - x^{2} + 3$

Step 1.2

Reorder $x$ and $- x^{2}$ .

$- x^{2} + x + 3$

Step 2

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

$a = - 1$

$b = 1$

$c = 3$

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

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

Step 4.2

Cancel the common factor of $1$ and $- 1$ .

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Step 4.2.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 4.2.2

Move the negative in front of the fraction.

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

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

$e = 3 - \frac{1^{2}}{4 \cdot - 1}$

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.

$e = 3 - \frac{1}{4 \cdot - 1}$

Step 5.2.1.2

Multiply $4$ by $- 1$ .

$e = 3 - \frac{1}{- 4}$

Step 5.2.1.3

Move the negative in front of the fraction.

$e = 3 - - \frac{1}{4}$

Step 5.2.1.4

Multiply $- - \frac{1}{4}$ .

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Step 5.2.1.4.1

Multiply $- 1$ by $- 1$ .

$e = 3 + 1 (\frac{1}{4})$

Step 5.2.1.4.2

Multiply $\frac{1}{4}$ by $1$ .

$e = 3 + \frac{1}{4}$

Step 5.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$e = 3 \cdot \frac{4}{4} + \frac{1}{4}$

Step 5.2.3

Combine $3$ and $\frac{4}{4}$ .

$e = \frac{3 \cdot 4}{4} + \frac{1}{4}$

Step 5.2.4

Combine the numerators over the common denominator.

$e = \frac{3 \cdot 4 + 1}{4}$

Step 5.2.5

Simplify the numerator.

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Step 5.2.5.1

Multiply $3$ by $4$ .

$e = \frac{12 + 1}{4}$

Step 5.2.5.2

Add $12$ and $1$ .

$e = \frac{13}{4}$

Step 6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - \frac{1}{2})}^{2} + \frac{13}{4}$ .

$- {(x - \frac{1}{2})}^{2} + \frac{13}{4}$