Algebra Examples

$x^{2} + 11 x + \frac{121}{4} = {(x + n)}^{2}$

Step 1

Rewrite the equation as ${(x + n)}^{2} = x^{2} + 11 x + \frac{121}{4}$ .

${(x + n)}^{2} = x^{2} + 11 x + \frac{121}{4}$

Step 2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x + n = \pm \sqrt{x^{2} + 11 x + \frac{121}{4}}$

Step 3

Simplify $\pm \sqrt{x^{2} + 11 x + \frac{121}{4}}$ .

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Step 3.1

Factor using the perfect square rule.

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Step 3.1.1

Rewrite $121$ as $11^{2}$ .

$x + n = \pm \sqrt{x^{2} + 11 x + \frac{11^{2}}{4}}$

Step 3.1.2

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

$x + n = \pm \sqrt{x^{2} + 11 x + \frac{11^{2}}{2^{2}}}$

Step 3.1.3

Rewrite $\frac{11^{2}}{2^{2}}$ as ${(\frac{11}{2})}^{2}$ .

$x + n = \pm \sqrt{x^{2} + 11 x + {(\frac{11}{2})}^{2}}$

Step 3.1.4

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

$11 x = 2 \cdot x \cdot \frac{11}{2}$

Step 3.1.5

Rewrite the polynomial.

$x + n = \pm \sqrt{x^{2} + 2 \cdot x \cdot \frac{11}{2} + {(\frac{11}{2})}^{2}}$

Step 3.1.6

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = \frac{11}{2}$ .

$x + n = \pm \sqrt{{(x + \frac{11}{2})}^{2}}$

Step 3.2

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

$x + n = \pm \sqrt{{(x \cdot \frac{2}{2} + \frac{11}{2})}^{2}}$

Step 3.3

Simplify terms.

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Step 3.3.1

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

$x + n = \pm \sqrt{{(\frac{x \cdot 2}{2} + \frac{11}{2})}^{2}}$

Step 3.3.2

Combine the numerators over the common denominator.

$x + n = \pm \sqrt{{(\frac{x \cdot 2 + 11}{2})}^{2}}$

Step 3.4

Move $2$ to the left of $x$ .

$x + n = \pm \sqrt{{(\frac{2 x + 11}{2})}^{2}}$

Step 3.5

Simplify the expression.

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Step 3.5.1

Apply the product rule to $\frac{2 x + 11}{2}$ .

$x + n = \pm \sqrt{\frac{{(2 x + 11)}^{2}}{2^{2}}}$

Step 3.5.2

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

$x + n = \pm \sqrt{\frac{{(2 x + 11)}^{2}}{4}}$

Step 3.5.3

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

$x + n = \pm \sqrt{\frac{{(2 x + 11)}^{2}}{2^{2}}}$

Step 3.6

Rewrite $\frac{{(2 x + 11)}^{2}}{2^{2}}$ as ${(\frac{2 x + 11}{2})}^{2}$ .

$x + n = \pm \sqrt{{(\frac{2 x + 11}{2})}^{2}}$

Step 3.7

Pull terms out from under the radical, assuming positive real numbers.

$x + n = \pm \frac{2 x + 11}{2}$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.1

First, use the positive value of the $\pm$ to find the first solution.

$x + n = \frac{2 x + 11}{2}$

Step 4.2

Move all terms not containing $n$ to the right side of the equation.

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Step 4.2.1

Subtract $x$ from both sides of the equation.

$n = \frac{2 x + 11}{2} - x$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Split the fraction $\frac{2 x + 11}{2}$ into two fractions.

$n = \frac{2 x}{2} + \frac{11}{2} - x$

Step 4.2.2.2

Cancel the common factor of $2$ .

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Step 4.2.2.2.1

Cancel the common factor.

$n = \frac{2 x}{2} + \frac{11}{2} - x$

Step 4.2.2.2.2

Divide $x$ by $1$ .

$n = x + \frac{11}{2} - x$

Step 4.2.3

Combine the opposite terms in $x + \frac{11}{2} - x$ .

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Step 4.2.3.1

Subtract $x$ from $x$ .

$n = 0 + \frac{11}{2}$

Step 4.2.3.2

Add $0$ and $\frac{11}{2}$ .

$n = \frac{11}{2}$

Step 4.3

Next, use the negative value of the $\pm$ to find the second solution.

$x + n = - \frac{2 x + 11}{2}$

Step 4.4

Move all terms not containing $n$ to the right side of the equation.

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Step 4.4.1

Subtract $x$ from both sides of the equation.

$n = - \frac{2 x + 11}{2} - x$

Step 4.4.2

Simplify each term.

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Step 4.4.2.1

Split the fraction $\frac{2 x + 11}{2}$ into two fractions.

$n = - (\frac{2 x}{2} + \frac{11}{2}) - x$

Step 4.4.2.2

Cancel the common factor of $2$ .

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Step 4.4.2.2.1

Cancel the common factor.

$n = - (\frac{2 x}{2} + \frac{11}{2}) - x$

Step 4.4.2.2.2

Divide $x$ by $1$ .

$n = - (x + \frac{11}{2}) - x$

Step 4.4.2.3

Apply the distributive property.

$n = - x - \frac{11}{2} - x$

Step 4.4.3

Subtract $x$ from $- x$ .

$n = - 2 x - \frac{11}{2}$

Step 4.5

The complete solution is the result of both the positive and negative portions of the solution.

$n = \frac{11}{2}$

$n = - 2 x - \frac{11}{2}$

$n = \frac{11}{2}$

$n = - 2 x - \frac{11}{2}$