Algebra Examples

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

Step 1

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

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Step 1.1

Rewrite.

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

Step 1.2

Simplify by adding zeros.

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

Step 1.3

Expand $(x + a^{2}) (x + b^{2})$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

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

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

Step 1.4

Multiply $x$ by $x$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Multiply $x$ by $x$ .

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

Step 2.3.1.2

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

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Step 2.3.1.2.1

Move $a$ .

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

Step 2.3.1.2.2

Multiply $a$ by $a$ .

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

Step 2.3.1.3

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

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Step 2.3.1.3.1

Move $b$ .

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

Step 2.3.1.3.2

Multiply $b$ by $b$ .

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

Step 2.3.2

Add $x a b$ and $a b x$ .

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Step 2.3.2.1

Move $x$ .

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

Step 2.3.2.2

Add $a b x$ and $a b x$ .

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

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.

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

Step 3.2

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 4

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

Factor $x$ out of $x b^{2} + a^{2} x - 2 a b x$ .

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Factor $x$ out of $x (b^{2}) + x a^{2}$ .

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

Step 5.5

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

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

Step 6

Factor using the perfect square rule.

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Step 6.1

Rearrange terms.

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

Step 6.2

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

$2 a b = 2 \cdot b \cdot a$

Step 6.3

Rewrite the polynomial.

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

Step 6.4

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

$x {(b - a)}^{2} = 0$

Step 7

Divide each term in $x {(b - a)}^{2} = 0$ by ${(b - a)}^{2}$ and simplify.

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Step 7.1

Divide each term in $x {(b - a)}^{2} = 0$ by ${(b - a)}^{2}$ .

$\frac{x {(b - a)}^{2}}{{(b - a)}^{2}} = \frac{0}{{(b - a)}^{2}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of ${(b - a)}^{2}$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{x {(b - a)}^{2}}{{(b - a)}^{2}} = \frac{0}{{(b - a)}^{2}}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{{(b - a)}^{2}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $0$ by ${(b - a)}^{2}$ .

$x = 0$