Algebra Examples

$(x + \frac{13}{2}) (x - \frac{13}{2}) = 0$

Step 1

Simplify the left side.

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Step 1.1

Simplify $(x + \frac{13}{2}) (x - \frac{13}{2})$ .

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Step 1.1.1

Expand $(x + \frac{13}{2}) (x - \frac{13}{2})$ using the FOIL Method.

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Step 1.1.1.1

Apply the distributive property.

$x (x - \frac{13}{2}) + \frac{13}{2} (x - \frac{13}{2}) = 0$

Step 1.1.1.2

Apply the distributive property.

$x \cdot x + x (- \frac{13}{2}) + \frac{13}{2} (x - \frac{13}{2}) = 0$

Step 1.1.1.3

Apply the distributive property.

$x \cdot x + x (- \frac{13}{2}) + \frac{13}{2} x + \frac{13}{2} (- \frac{13}{2}) = 0$

Step 1.1.2

Simplify terms.

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Step 1.1.2.1

Combine the opposite terms in $x \cdot x + x (- \frac{13}{2}) + \frac{13}{2} x + \frac{13}{2} (- \frac{13}{2})$ .

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Step 1.1.2.1.1

Reorder the factors in the terms $x (- \frac{13}{2})$ and $\frac{13}{2} x$ .

$x \cdot x - \frac{13}{2} x + \frac{13}{2} x + \frac{13}{2} (- \frac{13}{2}) = 0$

Step 1.1.2.1.2

Add $- \frac{13}{2} x$ and $\frac{13}{2} x$ .

$x \cdot x + 0 + \frac{13}{2} (- \frac{13}{2}) = 0$

Step 1.1.2.1.3

Add $x \cdot x$ and $0$ .

$x \cdot x + \frac{13}{2} (- \frac{13}{2}) = 0$

Step 1.1.2.2

Simplify each term.

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Step 1.1.2.2.1

Multiply $x$ by $x$ .

$x^{2} + \frac{13}{2} (- \frac{13}{2}) = 0$

Step 1.1.2.2.2

Multiply $\frac{13}{2} (- \frac{13}{2})$ .

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Step 1.1.2.2.2.1

Multiply $\frac{13}{2}$ by $\frac{13}{2}$ .

$x^{2} - \frac{13 \cdot 13}{2 \cdot 2} = 0$

Step 1.1.2.2.2.2

Multiply $13$ by $13$ .

$x^{2} - \frac{169}{2 \cdot 2} = 0$

Step 1.1.2.2.2.3

Multiply $2$ by $2$ .

$x^{2} - \frac{169}{4} = 0$

Step 2

Multiply through by the least common denominator $4$ , then simplify.

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Step 2.1

Apply the distributive property.

$4 x^{2} + 4 (- \frac{169}{4}) = 0$

Step 2.2

Cancel the common factor of $4$ .

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Step 2.2.1

Move the leading negative in $- \frac{169}{4}$ into the numerator.

$4 x^{2} + 4 (\frac{- 169}{4}) = 0$

Step 2.2.2

Cancel the common factor.

$4 x^{2} + 4 (\frac{- 169}{4}) = 0$

Step 2.2.3

Rewrite the expression.

$4 x^{2} - 169 = 0$

Step 3

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values $a = 4$ , $b = 0$ , and $c = - 169$ into the quadratic formula and solve for $x$ .

$\frac{0 \pm \sqrt{0^{2} - 4 \cdot (4 \cdot - 169)}}{2 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raising $0$ to any positive power yields $0$ .

$x = \frac{0 \pm \sqrt{0 - 4 \cdot 4 \cdot - 169}}{2 \cdot 4}$

Step 5.1.2

Multiply $- 4 \cdot 4 \cdot - 169$ .

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Step 5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{0 \pm \sqrt{0 - 16 \cdot - 169}}{2 \cdot 4}$

Step 5.1.2.2

Multiply $- 16$ by $- 169$ .

$x = \frac{0 \pm \sqrt{0 + 2704}}{2 \cdot 4}$

Step 5.1.3

Add $0$ and $2704$ .

$x = \frac{0 \pm \sqrt{2704}}{2 \cdot 4}$

Step 5.1.4

Rewrite $2704$ as $52^{2}$ .

$x = \frac{0 \pm \sqrt{52^{2}}}{2 \cdot 4}$

Step 5.1.5

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

$x = \frac{0 \pm 52}{2 \cdot 4}$

Step 5.2

Multiply $2$ by $4$ .

$x = \frac{0 \pm 52}{8}$

Step 5.3

Simplify $\frac{0 \pm 52}{8}$ .

$x = \frac{\pm 13}{2}$

Step 6

The final answer is the combination of both solutions.

$x = \frac{13}{2}, - \frac{13}{2}$