Algebra Examples

$(x + a) (x - a) = x^{2} - 16$

Step 1

Simplify $(x + a) (x - a)$ .

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Step 1.1

Expand $(x + a) (x - a)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$x (x - a) + a (x - a) = x^{2} - 16$

Step 1.1.2

Apply the distributive property.

$x \cdot x + x (- a) + a (x - a) = x^{2} - 16$

Step 1.1.3

Apply the distributive property.

$x \cdot x + x (- a) + a x + a (- a) = x^{2} - 16$

Step 1.2

Simplify terms.

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Step 1.2.1

Combine the opposite terms in $x \cdot x + x (- a) + a x + a (- a)$ .

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Step 1.2.1.1

Reorder the factors in the terms $x (- a)$ and $a x$ .

$x \cdot x - a x + a x + a (- a) = x^{2} - 16$

Step 1.2.1.2

Add $- a x$ and $a x$ .

$x \cdot x + 0 + a (- a) = x^{2} - 16$

Step 1.2.1.3

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

$x \cdot x + a (- a) = x^{2} - 16$

Step 1.2.2

Simplify each term.

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Step 1.2.2.1

Multiply $x$ by $x$ .

$x^{2} + a (- a) = x^{2} - 16$

Step 1.2.2.2

Rewrite using the commutative property of multiplication.

$x^{2} - a \cdot a = x^{2} - 16$

Step 1.2.2.3

Multiply $a$ by $a$ by adding the exponents.

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Step 1.2.2.3.1

Move $a$ .

$x^{2} - (a \cdot a) = x^{2} - 16$

Step 1.2.2.3.2

Multiply $a$ by $a$ .

$x^{2} - a^{2} = x^{2} - 16$

Step 2

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

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Step 2.1

Subtract $x^{2}$ from both sides of the equation.

$- a^{2} = x^{2} - 16 - x^{2}$

Step 2.2

Combine the opposite terms in $x^{2} - 16 - x^{2}$ .

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Step 2.2.1

Subtract $x^{2}$ from $x^{2}$ .

$- a^{2} = 0 - 16$

Step 2.2.2

Subtract $16$ from $0$ .

$- a^{2} = - 16$

Step 3

Divide each term in $- a^{2} = - 16$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- a^{2} = - 16$ by $- 1$ .

$\frac{- a^{2}}{- 1} = \frac{- 16}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{a^{2}}{1} = \frac{- 16}{- 1}$

Step 3.2.2

Divide $a^{2}$ by $1$ .

$a^{2} = \frac{- 16}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $- 16$ by $- 1$ .

$a^{2} = 16$

Step 4

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

$a = \pm \sqrt{16}$

Step 5

Simplify $\pm \sqrt{16}$ .

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Step 5.1

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

$a = \pm \sqrt{4^{2}}$

Step 5.2

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

$a = \pm 4$

Step 6

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

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Step 6.1

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

$a = 4$

Step 6.2

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

$a = - 4$

Step 6.3

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

$a = 4, - 4$