Basic Math Examples

$a^{2} + {(2 b + 6)}^{2} = {(2 c + 4)}^{2}$

Step 1

Simplify ${(2 c + 4)}^{2}$ .

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Step 1.1

Rewrite.

$a^{2} + {(2 b + 6)}^{2} = 0 + 0 + {(2 c + 4)}^{2}$

Step 1.2

Rewrite ${(2 c + 4)}^{2}$ as $(2 c + 4) (2 c + 4)$ .

$a^{2} + {(2 b + 6)}^{2} = (2 c + 4) (2 c + 4)$

Step 1.3

Expand $(2 c + 4) (2 c + 4)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$a^{2} + {(2 b + 6)}^{2} = 2 c (2 c + 4) + 4 (2 c + 4)$

Step 1.3.2

Apply the distributive property.

$a^{2} + {(2 b + 6)}^{2} = 2 c (2 c) + 2 c \cdot 4 + 4 (2 c + 4)$

Step 1.3.3

Apply the distributive property.

$a^{2} + {(2 b + 6)}^{2} = 2 c (2 c) + 2 c \cdot 4 + 4 (2 c) + 4 \cdot 4$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Rewrite using the commutative property of multiplication.

$a^{2} + {(2 b + 6)}^{2} = 2 \cdot 2 c \cdot c + 2 c \cdot 4 + 4 (2 c) + 4 \cdot 4$

Step 1.4.1.2

Multiply $c$ by $c$ by adding the exponents.

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Step 1.4.1.2.1

Move $c$ .

$a^{2} + {(2 b + 6)}^{2} = 2 \cdot 2 (c \cdot c) + 2 c \cdot 4 + 4 (2 c) + 4 \cdot 4$

Step 1.4.1.2.2

Multiply $c$ by $c$ .

$a^{2} + {(2 b + 6)}^{2} = 2 \cdot 2 c^{2} + 2 c \cdot 4 + 4 (2 c) + 4 \cdot 4$

Step 1.4.1.3

Multiply $2$ by $2$ .

$a^{2} + {(2 b + 6)}^{2} = 4 c^{2} + 2 c \cdot 4 + 4 (2 c) + 4 \cdot 4$

Step 1.4.1.4

Multiply $4$ by $2$ .

$a^{2} + {(2 b + 6)}^{2} = 4 c^{2} + 8 c + 4 (2 c) + 4 \cdot 4$

Step 1.4.1.5

Multiply $2$ by $4$ .

$a^{2} + {(2 b + 6)}^{2} = 4 c^{2} + 8 c + 8 c + 4 \cdot 4$

Step 1.4.1.6

Multiply $4$ by $4$ .

$a^{2} + {(2 b + 6)}^{2} = 4 c^{2} + 8 c + 8 c + 16$

Step 1.4.2

Add $8 c$ and $8 c$ .

$a^{2} + {(2 b + 6)}^{2} = 4 c^{2} + 16 c + 16$

Step 2

Subtract ${(2 b + 6)}^{2}$ from both sides of the equation.

$a^{2} = 4 c^{2} + 16 c + 16 - {(2 b + 6)}^{2}$

Step 3

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

$a = \pm \sqrt{4 c^{2} + 16 c + 16 - {(2 b + 6)}^{2}}$

Step 4

Simplify $\pm \sqrt{4 c^{2} + 16 c + 16 - {(2 b + 6)}^{2}}$ .

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Step 4.1

Factor using the perfect square rule.

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Step 4.1.1

Rewrite $4 c^{2}$ as ${(2 c)}^{2}$ .

$a = \pm \sqrt{{(2 c)}^{2} + 16 c + 16 - {(2 b + 6)}^{2}}$

Step 4.1.2

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

$a = \pm \sqrt{{(2 c)}^{2} + 16 c + 4^{2} - {(2 b + 6)}^{2}}$

Step 4.1.3

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

$16 c = 2 \cdot (2 c) \cdot 4$

Step 4.1.4

Rewrite the polynomial.

$a = \pm \sqrt{{(2 c)}^{2} + 2 \cdot (2 c) \cdot 4 + 4^{2} - {(2 b + 6)}^{2}}$

Step 4.1.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 2 c$ and $b = 4$ .

$a = \pm \sqrt{{(2 c + 4)}^{2} - {(2 b + 6)}^{2}}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 c + 4$ and $b = 2 b + 6$ .

$a = \pm \sqrt{(2 c + 4 + 2 b + 6) (2 c + 4 - (2 b + 6))}$

Step 4.3

Simplify.

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Step 4.3.1

Add $4$ and $6$ .

$a = \pm \sqrt{(2 c + 2 b + 10) (2 c + 4 - (2 b + 6))}$

Step 4.3.2

Factor $2$ out of $2 c + 2 b + 10$ .

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Step 4.3.2.1

Factor $2$ out of $2 c$ .

$a = \pm \sqrt{(2 (c) + 2 b + 10) (2 c + 4 - (2 b + 6))}$

Step 4.3.2.2

Factor $2$ out of $2 b$ .

$a = \pm \sqrt{(2 (c) + 2 (b) + 10) (2 c + 4 - (2 b + 6))}$

Step 4.3.2.3

Factor $2$ out of $10$ .

$a = \pm \sqrt{(2 (c) + 2 b + 2 \cdot 5) (2 c + 4 - (2 b + 6))}$

Step 4.3.2.4

Factor $2$ out of $2 (c) + 2 b$ .

$a = \pm \sqrt{(2 (c + b) + 2 \cdot 5) (2 c + 4 - (2 b + 6))}$

Step 4.3.2.5

Factor $2$ out of $2 (c + b) + 2 \cdot 5$ .

$a = \pm \sqrt{2 (c + b + 5) (2 c + 4 - (2 b + 6))}$

Step 4.3.3

Apply the distributive property.

$a = \pm \sqrt{2 (c + b + 5) (2 c + 4 - (2 b) - 1 \cdot 6)}$

Step 4.3.4

Multiply $2$ by $- 1$ .

$a = \pm \sqrt{2 (c + b + 5) (2 c + 4 - 2 b - 1 \cdot 6)}$

Step 4.3.5

Multiply $- 1$ by $6$ .

$a = \pm \sqrt{2 (c + b + 5) (2 c + 4 - 2 b - 6)}$

Step 4.3.6

Subtract $6$ from $4$ .

$a = \pm \sqrt{2 (c + b + 5) (2 c - 2 b - 2)}$

Step 4.3.7

Factor $2$ out of $2 c - 2 b - 2$ .

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Step 4.3.7.1

Factor $2$ out of $2 c$ .

$a = \pm \sqrt{2 (c + b + 5) (2 (c) - 2 b - 2)}$

Step 4.3.7.2

Factor $2$ out of $- 2 b$ .

$a = \pm \sqrt{2 (c + b + 5) (2 (c) + 2 (- b) - 2)}$

Step 4.3.7.3

Factor $2$ out of $- 2$ .

$a = \pm \sqrt{2 (c + b + 5) (2 (c) + 2 (- b) + 2 (- 1))}$

Step 4.3.7.4

Factor $2$ out of $2 (c) + 2 (- b)$ .

$a = \pm \sqrt{2 (c + b + 5) (2 (c - b) + 2 (- 1))}$

Step 4.3.7.5

Factor $2$ out of $2 (c - b) + 2 (- 1)$ .

$a = \pm \sqrt{2 (c + b + 5) (2 (c - b - 1))}$

$a = \pm \sqrt{2 (c + b + 5) \cdot 2 (c - b - 1)}$

Step 4.3.8

Multiply $2$ by $2$ .

$a = \pm \sqrt{4 (c + b + 5) (c - b - 1)}$

Step 4.4

Rewrite $4 (c + b + 5) (c - b - 1)$ as $2^{2} ((c + b + 5) (c - b - 1))$ .

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Step 4.4.1

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

$a = \pm \sqrt{2^{2} (c + b + 5) (c - b - 1)}$

Step 4.4.2

Add parentheses.

$a = \pm \sqrt{2^{2} ((c + b + 5) (c - b - 1))}$

Step 4.5

Pull terms out from under the radical.

$a = \pm 2 \sqrt{(c + b + 5) (c - b - 1)}$

Step 5

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

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Step 5.1

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

$a = 2 \sqrt{(c + b + 5) (c - b - 1)}$

Step 5.2

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

$a = - 2 \sqrt{(c + b + 5) (c - b - 1)}$

Step 5.3

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

$a = 2 \sqrt{(c + b + 5) (c - b - 1)}$

$a = - 2 \sqrt{(c + b + 5) (c - b - 1)}$

$a = 2 \sqrt{(c + b + 5) (c - b - 1)}$

$a = - 2 \sqrt{(c + b + 5) (c - b - 1)}$