Basic Math Examples

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

Step 1

Simplify $(a + b) \cdot 2$ .

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Step 1.1

Rewrite.

$a^{2} + b^{2} + 2 a b = 0 + 0 + (a + b) \cdot 2$

Step 1.2

Simplify by multiplying through.

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Step 1.2.1

Apply the distributive property.

$a^{2} + b^{2} + 2 a b = a \cdot 2 + b \cdot 2$

Step 1.2.2

Reorder.

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Step 1.2.2.1

Move $2$ to the left of $a$ .

$a^{2} + b^{2} + 2 a b = 2 \cdot a + b \cdot 2$

Step 1.2.2.2

Move $2$ to the left of $b$ .

$a^{2} + b^{2} + 2 a b = 2 \cdot a + 2 \cdot b$

Step 1.3

Multiply $2$ by $b$ .

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

Step 2

Subtract $2 a$ from both sides of the equation.

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

Step 3

Subtract $2 b$ from both sides of the equation.

$a^{2} + b^{2} + 2 a b - 2 a - 2 b = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = 1$ , $b = 2 b - 2$ , and $c = b^{2} - 2 b$ into the quadratic formula and solve for $a$ .

$\frac{- (2 b - 2) \pm \sqrt{{(2 b - 2)}^{2} - 4 \cdot (1 \cdot (b^{2} - 2 b))}}{2 \cdot 1}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Apply the distributive property.

$a = \frac{- (2 b) + 2 \pm \sqrt{{(2 b - 2)}^{2} - 4 \cdot 1 \cdot (b^{2} - 2 b)}}{2 \cdot 1}$

Step 6.1.2

Multiply $2$ by $- 1$ .

$a = \frac{- 2 b + 2 \pm \sqrt{{(2 b - 2)}^{2} - 4 \cdot 1 \cdot (b^{2} - 2 b)}}{2 \cdot 1}$

Step 6.1.3

Multiply $- 1$ by $- 2$ .

$a = \frac{- 2 b + 2 \pm \sqrt{{(2 b - 2)}^{2} - 4 \cdot 1 \cdot (b^{2} - 2 b)}}{2 \cdot 1}$

Step 6.1.4

Add parentheses.

$a = \frac{- 2 b + 2 \pm \sqrt{{(2 b - 2)}^{2} - 4 \cdot (1 \cdot (b^{2} - 2 b))}}{2 \cdot 1}$

Step 6.1.5

Let $u = 1 \cdot (b^{2} - 2 b)$ . Substitute $u$ for all occurrences of $1 \cdot (b^{2} - 2 b)$ .

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Step 6.1.5.1

Rewrite ${(2 b - 2)}^{2}$ as $(2 b - 2) (2 b - 2)$ .

$a = \frac{- 2 b + 2 \pm \sqrt{(2 b - 2) (2 b - 2) - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.2

Expand $(2 b - 2) (2 b - 2)$ using the FOIL Method.

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Step 6.1.5.2.1

Apply the distributive property.

$a = \frac{- 2 b + 2 \pm \sqrt{2 b (2 b - 2) - 2 (2 b - 2) - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.2.2

Apply the distributive property.

$a = \frac{- 2 b + 2 \pm \sqrt{2 b (2 b) + 2 b \cdot - 2 - 2 (2 b - 2) - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.2.3

Apply the distributive property.

$a = \frac{- 2 b + 2 \pm \sqrt{2 b (2 b) + 2 b \cdot - 2 - 2 (2 b) - 2 \cdot - 2 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3

Simplify and combine like terms.

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Step 6.1.5.3.1

Simplify each term.

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Step 6.1.5.3.1.1

Rewrite using the commutative property of multiplication.

$a = \frac{- 2 b + 2 \pm \sqrt{2 \cdot (2 b \cdot b) + 2 b \cdot - 2 - 2 (2 b) - 2 \cdot - 2 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3.1.2

Multiply $b$ by $b$ by adding the exponents.

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Step 6.1.5.3.1.2.1

Move $b$ .

$a = \frac{- 2 b + 2 \pm \sqrt{2 \cdot (2 (b \cdot b)) + 2 b \cdot - 2 - 2 (2 b) - 2 \cdot - 2 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3.1.2.2

Multiply $b$ by $b$ .

$a = \frac{- 2 b + 2 \pm \sqrt{2 \cdot (2 b^{2}) + 2 b \cdot - 2 - 2 (2 b) - 2 \cdot - 2 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3.1.3

Multiply $2$ by $2$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 b^{2} + 2 b \cdot - 2 - 2 (2 b) - 2 \cdot - 2 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3.1.4

Multiply $- 2$ by $2$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 b^{2} - 4 b - 2 (2 b) - 2 \cdot - 2 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3.1.5

Multiply $2$ by $- 2$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 b^{2} - 4 b - 4 b - 2 \cdot - 2 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3.1.6

Multiply $- 2$ by $- 2$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 b^{2} - 4 b - 4 b + 4 - 4 \cdot u}}{2 \cdot 1}$

Step 6.1.5.3.2

Subtract $4 b$ from $- 4 b$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 b^{2} - 8 b + 4 - 4 \cdot u}}{2 \cdot 1}$

$a = \frac{- 2 b + 2 \pm \sqrt{4 b^{2} - 8 b + 4 - 4 u}}{2 \cdot 1}$

Step 6.1.6

Factor $4$ out of $4 b^{2} - 8 b + 4 - 4 u$ .

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Step 6.1.6.1

Factor $4$ out of $4 b^{2}$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 (b^{2}) - 8 b + 4 - 4 u}}{2 \cdot 1}$

Step 6.1.6.2

Factor $4$ out of $- 8 b$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 (b^{2}) + 4 (- 2 b) + 4 - 4 u}}{2 \cdot 1}$

Step 6.1.6.3

Factor $4$ out of $4$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 (b^{2}) + 4 (- 2 b) + 4 (1) - 4 u}}{2 \cdot 1}$

Step 6.1.6.4

Factor $4$ out of $- 4 u$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 (b^{2}) + 4 (- 2 b) + 4 (1) + 4 (- u)}}{2 \cdot 1}$

Step 6.1.6.5

Factor $4$ out of $4 (b^{2}) + 4 (- 2 b)$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 (b^{2} - 2 b) + 4 (1) + 4 (- u)}}{2 \cdot 1}$

Step 6.1.6.6

Factor $4$ out of $4 (b^{2} - 2 b) + 4 (1)$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 (b^{2} - 2 b + 1) + 4 (- u)}}{2 \cdot 1}$

Step 6.1.6.7

Factor $4$ out of $4 (b^{2} - 2 b + 1) + 4 (- u)$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 (b^{2} - 2 b + 1 - u)}}{2 \cdot 1}$

Step 6.1.7

Replace all occurrences of $u$ with $1 \cdot (b^{2} - 2 b)$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 (b^{2} - 2 b + 1 - (1 \cdot (b^{2} - 2 b)))}}{2 \cdot 1}$

Step 6.1.8

Simplify.

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Step 6.1.8.1

Simplify each term.

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Step 6.1.8.1.1

Multiply $b^{2} - 2 b$ by $1$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 (b^{2} - 2 b + 1 - (b^{2} - 2 b))}}{2 \cdot 1}$

Step 6.1.8.1.2

Apply the distributive property.

$a = \frac{- 2 b + 2 \pm \sqrt{4 (b^{2} - 2 b + 1 - b^{2} - (- 2 b))}}{2 \cdot 1}$

Step 6.1.8.1.3

Multiply $- 2$ by $- 1$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 (b^{2} - 2 b + 1 - b^{2} + 2 b)}}{2 \cdot 1}$

Step 6.1.8.2

Combine the opposite terms in $b^{2} - 2 b + 1 - b^{2} + 2 b$ .

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Step 6.1.8.2.1

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

$a = \frac{- 2 b + 2 \pm \sqrt{4 (- 2 b + 1 + 0 + 2 b)}}{2 \cdot 1}$

Step 6.1.8.2.2

Add $- 2 b + 1$ and $0$ .

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

Step 6.1.8.2.3

Add $- 2 b$ and $2 b$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 (0 + 1)}}{2 \cdot 1}$

Step 6.1.8.2.4

Add $0$ and $1$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4 \cdot 1}}{2 \cdot 1}$

Step 6.1.9

Multiply $4$ by $1$ .

$a = \frac{- 2 b + 2 \pm \sqrt{4}}{2 \cdot 1}$

Step 6.1.10

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

$a = \frac{- 2 b + 2 \pm \sqrt{2^{2}}}{2 \cdot 1}$

Step 6.1.11

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

$a = \frac{- 2 b + 2 \pm 2}{2 \cdot 1}$

Step 6.2

Multiply $2$ by $1$ .

$a = \frac{- 2 b + 2 \pm 2}{2}$

Step 7

The final answer is the combination of both solutions.

$a = - b + 2$

$a = - b$