Algebra Examples

$x^{2} + a^{2} = {(a + x)}^{2} - a (a - 1)$

Step 1

Simplify ${(a + x)}^{2} - a (a - 1)$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(a + x)}^{2}$ as $(a + x) (a + x)$ .

$x^{2} + a^{2} = (a + x) (a + x) - a (a - 1)$

Step 1.1.2

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

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Step 1.1.2.1

Apply the distributive property.

$x^{2} + a^{2} = a (a + x) + x (a + x) - a (a - 1)$

Step 1.1.2.2

Apply the distributive property.

$x^{2} + a^{2} = a \cdot a + a x + x (a + x) - a (a - 1)$

Step 1.1.2.3

Apply the distributive property.

$x^{2} + a^{2} = a \cdot a + a x + x a + x \cdot x - a (a - 1)$

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $a$ by $a$ .

$x^{2} + a^{2} = a^{2} + a x + x a + x \cdot x - a (a - 1)$

Step 1.1.3.1.2

Multiply $x$ by $x$ .

$x^{2} + a^{2} = a^{2} + a x + x a + x^{2} - a (a - 1)$

Step 1.1.3.2

Add $a x$ and $x a$ .

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Step 1.1.3.2.1

Reorder $x$ and $a$ .

$x^{2} + a^{2} = a^{2} + a x + a x + x^{2} - a (a - 1)$

Step 1.1.3.2.2

Add $a x$ and $a x$ .

$x^{2} + a^{2} = a^{2} + 2 a x + x^{2} - a (a - 1)$

Step 1.1.4

Apply the distributive property.

$x^{2} + a^{2} = a^{2} + 2 a x + x^{2} - a \cdot a - a \cdot - 1$

Step 1.1.5

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

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Step 1.1.5.1

Move $a$ .

$x^{2} + a^{2} = a^{2} + 2 a x + x^{2} - (a \cdot a) - a \cdot - 1$

Step 1.1.5.2

Multiply $a$ by $a$ .

$x^{2} + a^{2} = a^{2} + 2 a x + x^{2} - a^{2} - a \cdot - 1$

Step 1.1.6

Multiply $- a \cdot - 1$ .

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Step 1.1.6.1

Multiply $- 1$ by $- 1$ .

$x^{2} + a^{2} = a^{2} + 2 a x + x^{2} - a^{2} + 1 a$

Step 1.1.6.2

Multiply $a$ by $1$ .

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

Step 1.2

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

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Step 1.2.1

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

$x^{2} + a^{2} = 2 a x + x^{2} + 0 + a$

Step 1.2.2

Add $2 a x + x^{2}$ and $0$ .

$x^{2} + a^{2} = 2 a x + x^{2} + a$

Step 2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 a x + x^{2} + a = x^{2} + a^{2}$

Step 3

Move all terms containing $x$ to the left side of the equation.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

$2 a x + a + 0 = a^{2}$

Step 3.2.2

Add $2 a x + a$ and $0$ .

$2 a x + a = a^{2}$

Step 4

Subtract $a$ from both sides of the equation.

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

Step 5

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

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Step 5.1

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

$\frac{2 a x}{2 a} = \frac{a^{2}}{2 a} + \frac{- a}{2 a}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{2 a x}{2 a} = \frac{a^{2}}{2 a} + \frac{- a}{2 a}$

Step 5.2.1.2

Rewrite the expression.

$\frac{a x}{a} = \frac{a^{2}}{2 a} + \frac{- a}{2 a}$

Step 5.2.2

Cancel the common factor of $a$ .

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Step 5.2.2.1

Cancel the common factor.

$\frac{a x}{a} = \frac{a^{2}}{2 a} + \frac{- a}{2 a}$

Step 5.2.2.2

Divide $x$ by $1$ .

$x = \frac{a^{2}}{2 a} + \frac{- a}{2 a}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Cancel the common factor of $a^{2}$ and $a$ .

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Step 5.3.1.1.1

Factor $a$ out of $a^{2}$ .

$x = \frac{a \cdot a}{2 a} + \frac{- a}{2 a}$

Step 5.3.1.1.2

Cancel the common factors.

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Step 5.3.1.1.2.1

Factor $a$ out of $2 a$ .

$x = \frac{a \cdot a}{a \cdot 2} + \frac{- a}{2 a}$

Step 5.3.1.1.2.2

Cancel the common factor.

$x = \frac{a \cdot a}{a \cdot 2} + \frac{- a}{2 a}$

Step 5.3.1.1.2.3

Rewrite the expression.

$x = \frac{a}{2} + \frac{- a}{2 a}$

Step 5.3.1.2

Cancel the common factor of $a$ .

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Step 5.3.1.2.1

Cancel the common factor.

$x = \frac{a}{2} + \frac{- a}{2 a}$

Step 5.3.1.2.2

Rewrite the expression.

$x = \frac{a}{2} + \frac{- 1}{2}$

Step 5.3.1.3

Move the negative in front of the fraction.

$x = \frac{a}{2} - \frac{1}{2}$