Algebra Examples

$y = {(x + \frac{3}{2})}^{2} - 1$

Step 1

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

$y = a x^{2} + b x + c$

Step 2

Simplify each term.

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Step 2.1

Rewrite ${(x + \frac{3}{2})}^{2}$ as $(x + \frac{3}{2}) (x + \frac{3}{2})$ .

$y = (x + \frac{3}{2}) (x + \frac{3}{2}) - 1$

Step 2.2

Expand $(x + \frac{3}{2}) (x + \frac{3}{2})$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$y = x (x + \frac{3}{2}) + \frac{3}{2} \cdot (x + \frac{3}{2}) - 1$

Step 2.2.2

Apply the distributive property.

$y = x \cdot x + x (\frac{3}{2}) + \frac{3}{2} \cdot (x + \frac{3}{2}) - 1$

Step 2.2.3

Apply the distributive property.

$y = x \cdot x + x (\frac{3}{2}) + \frac{3}{2} x + \frac{3}{2} \cdot \frac{3}{2} - 1$

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 $x$ .

$y = x^{2} + x (\frac{3}{2}) + \frac{3}{2} x + \frac{3}{2} \cdot \frac{3}{2} - 1$

Step 2.3.1.2

Combine $x$ and $\frac{3}{2}$ .

$y = x^{2} + \frac{x \cdot 3}{2} + \frac{3}{2} x + \frac{3}{2} \cdot \frac{3}{2} - 1$

Step 2.3.1.3

Move $3$ to the left of $x$ .

$y = x^{2} + \frac{3 \cdot x}{2} + \frac{3}{2} x + \frac{3}{2} \cdot \frac{3}{2} - 1$

Step 2.3.1.4

Combine $\frac{3}{2}$ and $x$ .

$y = x^{2} + \frac{3 x}{2} + \frac{3 x}{2} + \frac{3}{2} \cdot \frac{3}{2} - 1$

Step 2.3.1.5

Multiply $\frac{3}{2} \cdot \frac{3}{2}$ .

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Step 2.3.1.5.1

Multiply $\frac{3}{2}$ by $\frac{3}{2}$ .

$y = x^{2} + \frac{3 x}{2} + \frac{3 x}{2} + \frac{3 \cdot 3}{2 \cdot 2} - 1$

Step 2.3.1.5.2

Multiply $3$ by $3$ .

$y = x^{2} + \frac{3 x}{2} + \frac{3 x}{2} + \frac{9}{2 \cdot 2} - 1$

Step 2.3.1.5.3

Multiply $2$ by $2$ .

$y = x^{2} + \frac{3 x}{2} + \frac{3 x}{2} + \frac{9}{4} - 1$

Step 2.3.2

Add $\frac{3 x}{2}$ and $\frac{3 x}{2}$ .

$y = x^{2} + 2 (\frac{3 x}{2}) + \frac{9}{4} - 1$

Step 2.4

Cancel the common factor of $2$ .

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Step 2.4.1

Cancel the common factor.

$y = x^{2} + 2 (\frac{3 x}{2}) + \frac{9}{4} - 1$

Step 2.4.2

Rewrite the expression.

$y = x^{2} + 3 x + \frac{9}{4} - 1$

Step 3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = x^{2} + 3 x + \frac{9}{4} - 1 \cdot \frac{4}{4}$

Step 4

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

$y = x^{2} + 3 x + \frac{9}{4} + \frac{- 1 \cdot 4}{4}$

Step 5

Combine the numerators over the common denominator.

$y = x^{2} + 3 x + \frac{9 - 1 \cdot 4}{4}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $4$ .

$y = x^{2} + 3 x + \frac{9 - 4}{4}$

Step 6.2

Subtract $4$ from $9$ .

$y = x^{2} + 3 x + \frac{5}{4}$

Step 7