Algebra Examples

$4 x^{2} - 2 x + c$

Step 1

Move $- 2 x$ .

$4 x^{2} + c - 2 x$

Step 2

Factor out the $4$

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

Step 3

To find the $c$ value $c = {(\frac{b}{2})}^{2}$ , divide the coefficient of $x$ by $2$ and square the result.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

${(- (\frac{1}{2} \cdot \frac{1}{2}))}^{2}$

Step 3.2

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

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Step 3.2.1

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

${(- \frac{1}{2 \cdot 2})}^{2}$

Step 3.2.2

Multiply $2$ by $2$ .

${(- \frac{1}{4})}^{2}$

Step 3.3

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 3.3.1

Apply the product rule to $- \frac{1}{4}$ .

${(- 1)}^{2} {(\frac{1}{4})}^{2}$

Step 3.3.2

Apply the product rule to $\frac{1}{4}$ .

${(- 1)}^{2} \frac{1^{2}}{4^{2}}$

Step 3.4

Raise $- 1$ to the power of $2$ .

$1 \frac{1^{2}}{4^{2}}$

Step 3.5

Multiply $\frac{1^{2}}{4^{2}}$ by $1$ .

$\frac{1^{2}}{4^{2}}$

Step 3.6

One to any power is one.

$\frac{1}{4^{2}}$

Step 3.7

Raise $4$ to the power of $2$ .

$\frac{1}{16}$

Step 4

Add $\frac{1}{16}$ to get the perfect square trinomial.

$4 (x^{2} - \frac{x}{2} + \frac{c}{4} + \frac{1}{16})$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$4 x^{2} + 4 (- \frac{x}{2}) + 4 \frac{c}{4} + 4 (\frac{1}{16})$

Step 5.2

Simplify.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Move the leading negative in $- \frac{x}{2}$ into the numerator.

$4 x^{2} + 4 \frac{- x}{2} + 4 \frac{c}{4} + 4 (\frac{1}{16})$

Step 5.2.1.2

Factor $2$ out of $4$ .

$4 x^{2} + 2 (2) \frac{- x}{2} + 4 \frac{c}{4} + 4 (\frac{1}{16})$

Step 5.2.1.3

Cancel the common factor.

$4 x^{2} + 2 \cdot 2 \frac{- x}{2} + 4 \frac{c}{4} + 4 (\frac{1}{16})$

Step 5.2.1.4

Rewrite the expression.

$4 x^{2} + 2 (- x) + 4 \frac{c}{4} + 4 (\frac{1}{16})$

Step 5.2.2

Multiply $- 1$ by $2$ .

$4 x^{2} - 2 x + 4 \frac{c}{4} + 4 (\frac{1}{16})$

Step 5.2.3

Cancel the common factor of $4$ .

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Step 5.2.3.1

Cancel the common factor.

$4 x^{2} - 2 x + 4 \frac{c}{4} + 4 (\frac{1}{16})$

Step 5.2.3.2

Rewrite the expression.

$4 x^{2} - 2 x + c + 4 (\frac{1}{16})$

Step 5.2.4

Cancel the common factor of $4$ .

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Step 5.2.4.1

Factor $4$ out of $16$ .

$4 x^{2} - 2 x + c + 4 \frac{1}{4 (4)}$

Step 5.2.4.2

Cancel the common factor.

$4 x^{2} - 2 x + c + 4 \frac{1}{4 \cdot 4}$

Step 5.2.4.3

Rewrite the expression.

$4 x^{2} - 2 x + c + \frac{1}{4}$