Algebra Examples

$x^{2} + \frac{3}{2} x + \frac{9}{16}$

Step 1

Rewrite the middle term.

$x^{2} + 2 x \frac{3}{4} + 0 + \frac{9}{16}$

Step 2

Rearrange terms.

$x^{2} + 2 x \frac{3}{4} + \frac{9}{16} + 0$

Step 3

Factor first three terms by perfect square rule.

${(x + \frac{3}{4})}^{2} + 0$

Step 4

Rewrite ${(x + \frac{3}{4})}^{2}$ as $(x + \frac{3}{4}) (x + \frac{3}{4})$ .

$(x + \frac{3}{4}) (x + \frac{3}{4}) + 0$

Step 5

Expand $(x + \frac{3}{4}) (x + \frac{3}{4})$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$x (x + \frac{3}{4}) + \frac{3}{4} (x + \frac{3}{4}) + 0$

Step 5.2

Apply the distributive property.

$x \cdot x + x \frac{3}{4} + \frac{3}{4} (x + \frac{3}{4}) + 0$

Step 5.3

Apply the distributive property.

$x \cdot x + x \frac{3}{4} + \frac{3}{4} x + \frac{3}{4} \cdot \frac{3}{4} + 0$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $x$ by $x$ .

$x^{2} + x \frac{3}{4} + \frac{3}{4} x + \frac{3}{4} \cdot \frac{3}{4} + 0$

Step 6.1.2

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

$x^{2} + \frac{x \cdot 3}{4} + \frac{3}{4} x + \frac{3}{4} \cdot \frac{3}{4} + 0$

Step 6.1.3

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

$x^{2} + \frac{3 \cdot x}{4} + \frac{3}{4} x + \frac{3}{4} \cdot \frac{3}{4} + 0$

Step 6.1.4

Multiply $3$ by $x$ .

$x^{2} + \frac{3 x}{4} + \frac{3}{4} x + \frac{3}{4} \cdot \frac{3}{4} + 0$

Step 6.1.5

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

$x^{2} + \frac{3 x}{4} + \frac{3 x}{4} + \frac{3}{4} \cdot \frac{3}{4} + 0$

Step 6.1.6

Multiply $\frac{3}{4} \cdot \frac{3}{4}$ .

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Step 6.1.6.1

Multiply $\frac{3}{4}$ by $\frac{3}{4}$ .

$x^{2} + \frac{3 x}{4} + \frac{3 x}{4} + \frac{3 \cdot 3}{4 \cdot 4} + 0$

Step 6.1.6.2

Multiply $3$ by $3$ .

$x^{2} + \frac{3 x}{4} + \frac{3 x}{4} + \frac{9}{4 \cdot 4} + 0$

Step 6.1.6.3

Multiply $4$ by $4$ .

$x^{2} + \frac{3 x}{4} + \frac{3 x}{4} + \frac{9}{16} + 0$

Step 6.2

Add $\frac{3 x}{4}$ and $\frac{3 x}{4}$ .

$x^{2} + 2 \frac{3 x}{4} + \frac{9}{16} + 0$

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Factor $2$ out of $4$ .

$x^{2} + 2 \frac{3 x}{2 (2)} + \frac{9}{16} + 0$

Step 7.2

Cancel the common factor.

$x^{2} + 2 \frac{3 x}{2 \cdot 2} + \frac{9}{16} + 0$

Step 7.3

Rewrite the expression.

$x^{2} + \frac{3 x}{2} + \frac{9}{16} + 0$

Step 8

Add $x^{2} + \frac{3 x}{2} + \frac{9}{16}$ and $0$ .

$x^{2} + \frac{3 x}{2} + \frac{9}{16}$

Step 9

Factor using the perfect square rule.

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Step 9.1

Rewrite $9$ as $3^{2}$ .

$x^{2} + \frac{3 x}{2} + \frac{3^{2}}{16}$

Step 9.2

Rewrite $16$ as $4^{2}$ .

$x^{2} + \frac{3 x}{2} + \frac{3^{2}}{4^{2}}$

Step 9.3

Rewrite $\frac{3^{2}}{4^{2}}$ as ${(\frac{3}{4})}^{2}$ .

$x^{2} + \frac{3 x}{2} + {(\frac{3}{4})}^{2}$

Step 9.4

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$\frac{3 x}{2} = 2 \cdot x \cdot \frac{3}{4}$

Step 9.5

Rewrite the polynomial.

$x^{2} + 2 \cdot x \cdot \frac{3}{4} + {(\frac{3}{4})}^{2}$

Step 9.6

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = \frac{3}{4}$ .

${(x + \frac{3}{4})}^{2}$