Evaluate (a+b)^2=a^2+2ab+b^2

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

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

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

Simplify ${(a + b)}^{2}$ .

Rewrite.

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $a$ .

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

Multiply $b$ by $b$ .

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

Add $a b$ and $b a$ .

Reorder $b$ and $a$ .

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

Add $a b$ and $a b$ .

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

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

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

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

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

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

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

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

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

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

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

Subtract $2 a b$ from $2 a b$ .

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

Add $b^{2}$ and $0$ .

$b^{2} = b^{2}$

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

$| b | = | b |$

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

$b = b$

$b = - b$

$- b = b$

$- b = - b$

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

$b = b$

$b = - b$

Solve $b = b$ for $b$ .

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

Subtract $b$ from both sides of the equation.

$b - b = 0$

Subtract $b$ from $b$ .

$0 = 0$

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

Always true

Solve $b = - b$ for $b$ .

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

Add $b$ to both sides of the equation.

$b + b = 0$

Add $b$ and $b$ .

$2 b = 0$

Divide each term in $2 b = 0$ by $2$ and simplify.

Divide each term in $2 b = 0$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $b$ by $1$ .

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

Simplify the right side.

Divide $0$ by $2$ .

$b = 0$

List all of the solutions.

$b = 0$

The variable $a$ got canceled.

All real numbers

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$

Evaluate (a+b)^2=a^2+2ab+b^2

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