Algebra Examples

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

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 ${(x - \frac{1}{2})}^{2}$ as $(x - \frac{1}{2}) (x - \frac{1}{2})$ .

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

Step 2.2

Expand $(x - \frac{1}{2}) (x - \frac{1}{2})$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

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 $x$ .

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

Step 2.3.1.2

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

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

Step 2.3.1.3

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

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

Step 2.3.1.4

Multiply $- \frac{1}{2} (- \frac{1}{2})$ .

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Step 2.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 2.3.1.4.2

Multiply $\frac{1}{2}$ by $1$ .

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

Step 2.3.1.4.3

Multiply $\frac{1}{2}$ by $\frac{1}{2}$ .

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

Step 2.3.1.4.4

Multiply $2$ by $2$ .

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

Step 2.3.2

Subtract $\frac{x}{2}$ from $- \frac{x}{2}$ .

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

Step 2.4

Simplify each term.

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Step 2.4.1

Cancel the common factor of $2$ .

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Step 2.4.1.1

Factor $2$ out of $- 2$ .

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

Step 2.4.1.2

Cancel the common factor.

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

Step 2.4.1.3

Rewrite the expression.

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

Step 2.4.2

Rewrite $- 1 x$ as $- x$ .

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

Step 3

To write $\frac{3}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

Multiply $2$ by $2$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Multiply $3$ by $2$ .

$y = x^{2} - x + \frac{1 + 6}{4}$

Step 6.2

Add $1$ and $6$ .

$y = x^{2} - x + \frac{7}{4}$

Step 7