Algebra Examples

$y = 2 {(x - \frac{3}{2})}^{2} - \frac{5}{2}$

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

$y = 2 ((x - \frac{3}{2}) (x - \frac{3}{2})) - \frac{5}{2}$

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.

$y = 2 (x (x - \frac{3}{2}) - \frac{3}{2} \cdot (x - \frac{3}{2})) - \frac{5}{2}$

Step 2.2.2

Apply the distributive property.

$y = 2 (x \cdot x + x (- \frac{3}{2}) - \frac{3}{2} \cdot (x - \frac{3}{2})) - \frac{5}{2}$

Step 2.2.3

Apply the distributive property.

$y = 2 (x \cdot x + x (- \frac{3}{2}) - \frac{3}{2} x - \frac{3}{2} \cdot (- \frac{3}{2})) - \frac{5}{2}$

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 $x$ .

$y = 2 (x^{2} + x (- \frac{3}{2}) - \frac{3}{2} x - \frac{3}{2} \cdot (- \frac{3}{2})) - \frac{5}{2}$

Step 2.3.1.2

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

$y = 2 (x^{2} - \frac{x \cdot 3}{2} - \frac{3}{2} x - \frac{3}{2} \cdot (- \frac{3}{2})) - \frac{5}{2}$

Step 2.3.1.3

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

$y = 2 (x^{2} - \frac{3 \cdot x}{2} - \frac{3}{2} x - \frac{3}{2} \cdot (- \frac{3}{2})) - \frac{5}{2}$

Step 2.3.1.4

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

$y = 2 (x^{2} - \frac{3 x}{2} - \frac{x \cdot 3}{2} - \frac{3}{2} \cdot (- \frac{3}{2})) - \frac{5}{2}$

Step 2.3.1.5

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

$y = 2 (x^{2} - \frac{3 x}{2} - \frac{3 \cdot x}{2} - \frac{3}{2} \cdot (- \frac{3}{2})) - \frac{5}{2}$

Step 2.3.1.6

Multiply $- \frac{3}{2} (- \frac{3}{2})$ .

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Step 2.3.1.6.1

Multiply $- 1$ by $- 1$ .

$y = 2 (x^{2} - \frac{3 x}{2} - \frac{3 x}{2} + 1 (\frac{3}{2}) \cdot \frac{3}{2}) - \frac{5}{2}$

Step 2.3.1.6.2

Multiply $\frac{3}{2}$ by $1$ .

$y = 2 (x^{2} - \frac{3 x}{2} - \frac{3 x}{2} + \frac{3}{2} \cdot \frac{3}{2}) - \frac{5}{2}$

Step 2.3.1.6.3

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

$y = 2 (x^{2} - \frac{3 x}{2} - \frac{3 x}{2} + \frac{3 \cdot 3}{2 \cdot 2}) - \frac{5}{2}$

Step 2.3.1.6.4

Multiply $3$ by $3$ .

$y = 2 (x^{2} - \frac{3 x}{2} - \frac{3 x}{2} + \frac{9}{2 \cdot 2}) - \frac{5}{2}$

Step 2.3.1.6.5

Multiply $2$ by $2$ .

$y = 2 (x^{2} - \frac{3 x}{2} - \frac{3 x}{2} + \frac{9}{4}) - \frac{5}{2}$

Step 2.3.2

Subtract $\frac{3 x}{2}$ from $- \frac{3 x}{2}$ .

$y = 2 (x^{2} - 2 \frac{3 x}{2} + \frac{9}{4}) - \frac{5}{2}$

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 $- 2$ .

$y = 2 (x^{2} + 2 (- 1) (\frac{3 x}{2}) + \frac{9}{4}) - \frac{5}{2}$

Step 2.4.1.2

Cancel the common factor.

$y = 2 (x^{2} + 2 \cdot (- 1 \frac{3 x}{2}) + \frac{9}{4}) - \frac{5}{2}$

Step 2.4.1.3

Rewrite the expression.

$y = 2 (x^{2} - 1 (3 x) + \frac{9}{4}) - \frac{5}{2}$

Step 2.4.2

Multiply $3$ by $- 1$ .

$y = 2 (x^{2} - 3 x + \frac{9}{4}) - \frac{5}{2}$

Step 2.5

Apply the distributive property.

$y = 2 x^{2} + 2 (- 3 x) + 2 (\frac{9}{4}) - \frac{5}{2}$

Step 2.6

Simplify.

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Step 2.6.1

Multiply $- 3$ by $2$ .

$y = 2 x^{2} - 6 x + 2 (\frac{9}{4}) - \frac{5}{2}$

Step 2.6.2

Cancel the common factor of $2$ .

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Step 2.6.2.1

Factor $2$ out of $4$ .

$y = 2 x^{2} - 6 x + 2 (\frac{9}{2 (2)}) - \frac{5}{2}$

Step 2.6.2.2

Cancel the common factor.

$y = 2 x^{2} - 6 x + 2 (\frac{9}{2 \cdot 2}) - \frac{5}{2}$

Step 2.6.2.3

Rewrite the expression.

$y = 2 x^{2} - 6 x + \frac{9}{2} - \frac{5}{2}$

Step 3

Combine fractions.

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Step 3.1

Combine the numerators over the common denominator.

$y = 2 x^{2} - 6 x + \frac{9 - 5}{2}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Subtract $5$ from $9$ .

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

Step 3.2.2

Divide $4$ by $2$ .

$y = 2 x^{2} - 6 x + 2$

Step 4