Algebra Examples

$g (x) = \frac{1}{2} {(x + 2)}^{2} - 8$

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$g (x) = a x^{2} + b x + c$

Step 2

Simplify each term.

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Step 2.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$g (x) = \frac{1}{2} \cdot ((x + 2) (x + 2)) - 8$

Step 2.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$g (x) = \frac{1}{2} \cdot (x (x + 2) + 2 (x + 2)) - 8$

Step 2.2.2

Apply the distributive property.

$g (x) = \frac{1}{2} \cdot (x \cdot x + x \cdot 2 + 2 (x + 2)) - 8$

Step 2.2.3

Apply the distributive property.

$g (x) = \frac{1}{2} \cdot (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) - 8$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $x$ by $x$ .

$g (x) = \frac{1}{2} \cdot (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2) - 8$

Step 2.3.1.2

Move $2$ to the left of $x$ .

$g (x) = \frac{1}{2} \cdot (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2) - 8$

Step 2.3.1.3

Multiply $2$ by $2$ .

$g (x) = \frac{1}{2} \cdot (x^{2} + 2 x + 2 x + 4) - 8$

Step 2.3.2

Add $2 x$ and $2 x$ .

$g (x) = \frac{1}{2} \cdot (x^{2} + 4 x + 4) - 8$

Step 2.4

Apply the distributive property.

$g (x) = \frac{1}{2} x^{2} + \frac{1}{2} \cdot (4 x) + \frac{1}{2} \cdot 4 - 8$

Step 2.5

Simplify.

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Step 2.5.1

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

$g (x) = \frac{x^{2}}{2} + \frac{1}{2} \cdot (4 x) + \frac{1}{2} \cdot 4 - 8$

Step 2.5.2

Cancel the common factor of $2$ .

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Step 2.5.2.1

Factor $2$ out of $4 x$ .

$g (x) = \frac{x^{2}}{2} + \frac{1}{2} \cdot (2 (2 x)) + \frac{1}{2} \cdot 4 - 8$

Step 2.5.2.2

Cancel the common factor.

$g (x) = \frac{x^{2}}{2} + \frac{1}{2} \cdot (2 (2 x)) + \frac{1}{2} \cdot 4 - 8$

Step 2.5.2.3

Rewrite the expression.

$g (x) = \frac{x^{2}}{2} + 2 x + \frac{1}{2} \cdot 4 - 8$

Step 2.5.3

Cancel the common factor of $2$ .

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Step 2.5.3.1

Factor $2$ out of $4$ .

$g (x) = \frac{x^{2}}{2} + 2 x + \frac{1}{2} \cdot (2 (2)) - 8$

Step 2.5.3.2

Cancel the common factor.

$g (x) = \frac{x^{2}}{2} + 2 x + \frac{1}{2} \cdot (2 \cdot 2) - 8$

Step 2.5.3.3

Rewrite the expression.

$g (x) = \frac{x^{2}}{2} + 2 x + 2 - 8$

Step 3

Subtract $8$ from $2$ .

$g (x) = \frac{x^{2}}{2} + 2 x - 6$

Step 4

Reorder terms.

$g (x) = \frac{1}{2} x^{2} + 2 x - 6$