Linear Algebra Examples

{| p + q |}^{2} + {| p - q |}^{2} = 2 {| p |}^{2} + 2 {| q |}^{2}

${| p + q |}^{2} + {| p - q |}^{2} = 2 {| p |}^{2} + 2 {| q |}^{2}$

Step 1

Simplify each term.

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Step 1.1

Remove the absolute value in

${| p + q |}^{2}$ because exponentiations with even powers are always positive.

${(p + q)}^{2} + {| p - q |}^{2} = 2 {| p |}^{2} + 2 {| q |}^{2}$

Step 1.2

Remove the absolute value in

${| p - q |}^{2}$ because exponentiations with even powers are always positive.

${(p + q)}^{2} + {(p - q)}^{2} = 2 {| p |}^{2} + 2 {| q |}^{2}$

Step 2

Simplify each term.

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Step 2.1

Remove the absolute value in

${| p |}^{2}$ because exponentiations with even powers are always positive.

${(p + q)}^{2} + {(p - q)}^{2} = 2 p^{2} + 2 {| q |}^{2}$

Step 2.2

Remove the absolute value in

${| q |}^{2}$ because exponentiations with even powers are always positive.

${(p + q)}^{2} + {(p - q)}^{2} = 2 p^{2} + 2 q^{2}$

Step 3

Move all terms containing

$p$ to the left side of the equation.

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Step 3.1

Subtract

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

${(p + q)}^{2} + {(p - q)}^{2} - 2 p^{2} = 2 q^{2}$

Step 3.2

Simplify each term.

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Step 3.2.1

Rewrite

${(p + q)}^{2}$ as

$(p + q) (p + q)$ .

$(p + q) (p + q) + {(p - q)}^{2} - 2 p^{2} = 2 q^{2}$

Step 3.2.2

Expand

$(p + q) (p + q)$ using the FOIL Method.

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Step 3.2.2.1

Apply the distributive property.

$p (p + q) + q (p + q) + {(p - q)}^{2} - 2 p^{2} = 2 q^{2}$

Step 3.2.2.2

Apply the distributive property.

$p \cdot p + p q + q (p + q) + {(p - q)}^{2} - 2 p^{2} = 2 q^{2}$

Step 3.2.2.3

Apply the distributive property.

$p \cdot p + p q + q p + q \cdot q + {(p - q)}^{2} - 2 p^{2} = 2 q^{2}$

Step 3.2.3

Simplify and combine like terms.

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Step 3.2.3.1

Simplify each term.

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Step 3.2.3.1.1

Multiply

$p$ by

$p$ .

$p^{2} + p q + q p + q \cdot q + {(p - q)}^{2} - 2 p^{2} = 2 q^{2}$

Step 3.2.3.1.2

Multiply

$q$ by

$q$ .

$p^{2} + p q + q p + q^{2} + {(p - q)}^{2} - 2 p^{2} = 2 q^{2}$

Step 3.2.3.2

Add

$p q$ and

$q p$ .

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Step 3.2.3.2.1

Reorder

$q$ and

$p$ .

$p^{2} + p q + p q + q^{2} + {(p - q)}^{2} - 2 p^{2} = 2 q^{2}$

Step 3.2.3.2.2

Add

$p q$ and

$p q$ .

$p^{2} + 2 p q + q^{2} + {(p - q)}^{2} - 2 p^{2} = 2 q^{2}$

Step 3.2.4

Rewrite

${(p - q)}^{2}$ as

$(p - q) (p - q)$ .

$p^{2} + 2 p q + q^{2} + (p - q) (p - q) - 2 p^{2} = 2 q^{2}$

Step 3.2.5

Expand

$(p - q) (p - q)$ using the FOIL Method.

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Step 3.2.5.1

Apply the distributive property.

$p^{2} + 2 p q + q^{2} + p (p - q) - q (p - q) - 2 p^{2} = 2 q^{2}$

Step 3.2.5.2

Apply the distributive property.

$p^{2} + 2 p q + q^{2} + p \cdot p + p (- q) - q (p - q) - 2 p^{2} = 2 q^{2}$

Step 3.2.5.3

Apply the distributive property.

$p^{2} + 2 p q + q^{2} + p \cdot p + p (- q) - q p - q (- q) - 2 p^{2} = 2 q^{2}$

Step 3.2.6

Simplify and combine like terms.

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Step 3.2.6.1

Simplify each term.

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Step 3.2.6.1.1

Multiply

$p$ by

$p$ .

$p^{2} + 2 p q + q^{2} + p^{2} + p (- q) - q p - q (- q) - 2 p^{2} = 2 q^{2}$

Step 3.2.6.1.2

Rewrite using the commutative property of multiplication.

$p^{2} + 2 p q + q^{2} + p^{2} - p q - q p - q (- q) - 2 p^{2} = 2 q^{2}$

Step 3.2.6.1.3

Rewrite using the commutative property of multiplication.

$p^{2} + 2 p q + q^{2} + p^{2} - p q - q p - 1 \cdot - 1 q \cdot q - 2 p^{2} = 2 q^{2}$

Step 3.2.6.1.4

Multiply

$q$ by

$q$ by adding the exponents.

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Step 3.2.6.1.4.1

Move

$q$ .

$p^{2} + 2 p q + q^{2} + p^{2} - p q - q p - 1 \cdot - 1 (q \cdot q) - 2 p^{2} = 2 q^{2}$

Step 3.2.6.1.4.2

Multiply

$q$ by

$q$ .

$p^{2} + 2 p q + q^{2} + p^{2} - p q - q p - 1 \cdot - 1 q^{2} - 2 p^{2} = 2 q^{2}$

Step 3.2.6.1.5

Multiply

$- 1$ by

$- 1$ .

$p^{2} + 2 p q + q^{2} + p^{2} - p q - q p + 1 q^{2} - 2 p^{2} = 2 q^{2}$

Step 3.2.6.1.6

Multiply

$q^{2}$ by

$1$ .

$p^{2} + 2 p q + q^{2} + p^{2} - p q - q p + q^{2} - 2 p^{2} = 2 q^{2}$

Step 3.2.6.2

Subtract

$q p$ from

$- p q$ .

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Step 3.2.6.2.1

Move

$q$ .

$p^{2} + 2 p q + q^{2} + p^{2} - p q - 1 p q + q^{2} - 2 p^{2} = 2 q^{2}$

Step 3.2.6.2.2

Subtract

$p q$ from

$- p q$ .

$p^{2} + 2 p q + q^{2} + p^{2} - 2 p q + q^{2} - 2 p^{2} = 2 q^{2}$

Step 3.3

Combine the opposite terms in

$p^{2} + 2 p q + q^{2} + p^{2} - 2 p q + q^{2} - 2 p^{2}$ .

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Step 3.3.1

Subtract

$2 p q$ from

$2 p q$ .

$p^{2} + q^{2} + p^{2} + 0 + q^{2} - 2 p^{2} = 2 q^{2}$

Step 3.3.2

Add

$p^{2} + q^{2} + p^{2}$ and

$0$ .

$p^{2} + q^{2} + p^{2} + q^{2} - 2 p^{2} = 2 q^{2}$

Step 3.4

Add

$p^{2}$ and

$p^{2}$ .

$2 p^{2} + q^{2} + q^{2} - 2 p^{2} = 2 q^{2}$

Step 3.5

Combine the opposite terms in

$2 p^{2} + q^{2} + q^{2} - 2 p^{2}$ .

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Step 3.5.1

Subtract

$2 p^{2}$ from

$2 p^{2}$ .

$q^{2} + q^{2} + 0 = 2 q^{2}$

Step 3.5.2

Add

$q^{2} + q^{2}$ and

$0$ .

$q^{2} + q^{2} = 2 q^{2}$

Step 3.6

Add

$q^{2}$ and

$q^{2}$ .

$2 q^{2} = 2 q^{2}$

Step 4

Divide each term in

$2 q^{2} = 2 q^{2}$ by

$2$ and simplify.

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Step 4.1

Divide each term in

$2 q^{2} = 2 q^{2}$ by

$2$ .

$\frac{2 q^{2}}{2} = \frac{2 q^{2}}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of

$2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 q^{2}}{2} = \frac{2 q^{2}}{2}$

Step 4.2.1.2

Divide

$q^{2}$ by

$1$ .

$q^{2} = \frac{2 q^{2}}{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of

$2$ .

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Step 4.3.1.1

Cancel the common factor.

$q^{2} = \frac{2 q^{2}}{2}$

Step 4.3.1.2

Divide

$q^{2}$ by

$1$ .

$q^{2} = q^{2}$

Step 5

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| q | = | q |$

Step 6

Solve for

$q$ .

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Step 6.1

Rewrite the absolute value equation as four equations without absolute value bars.

$q = q$

$q = - q$

$- q = q$

$- q = - q$

Step 6.2

After simplifying, there are only two unique equations to be solved.

$q = q$

$q = - q$

Step 6.3

Solve

$q = q$ for

$q$ .

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Step 6.3.1

Move all terms containing

$q$ to the left side of the equation.

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Step 6.3.1.1

Subtract

$q$ from both sides of the equation.

$q - q = 0$

Step 6.3.1.2

Subtract

$q$ from

$q$ .

$0 = 0$

Step 6.3.2

Since

$0 = 0$ , the equation will always be true.

Always true

Step 6.4

Solve

$q = - q$ for

$q$ .

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Step 6.4.1

Move all terms containing

$q$ to the left side of the equation.

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Step 6.4.1.1

Add

$q$ to both sides of the equation.

$q + q = 0$

Step 6.4.1.2

Add

$q$ and

$q$ .

$2 q = 0$

Step 6.4.2

Divide each term in

$2 q = 0$ by

$2$ and simplify.

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Step 6.4.2.1

Divide each term in

$2 q = 0$ by

$2$ .

$\frac{2 q}{2} = \frac{0}{2}$

Step 6.4.2.2

Simplify the left side.

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Step 6.4.2.2.1

Cancel the common factor of

$2$ .

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Step 6.4.2.2.1.1

Cancel the common factor.

$\frac{2 q}{2} = \frac{0}{2}$

Step 6.4.2.2.1.2

Divide

$q$ by

$1$ .

$q = \frac{0}{2}$

Step 6.4.2.3

Simplify the right side.

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Step 6.4.2.3.1

Divide

$0$ by

$2$ .

$q = 0$

Step 6.5

List all of the solutions.

$q = 0$

Step 7

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 8