Finite Math Examples

$x^{2} + {(x + 10)}^{2} = {(x + 20)}^{2}$

Step 1

Subtract ${(x + 20)}^{2}$ from both sides of the equation.

$x^{2} + {(x + 10)}^{2} - {(x + 20)}^{2} = 0$

Step 2

Simplify $x^{2} + {(x + 10)}^{2} - {(x + 20)}^{2}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(x + 10)}^{2}$ as $(x + 10) (x + 10)$ .

$x^{2} + (x + 10) (x + 10) - {(x + 20)}^{2} = 0$

Step 2.1.2

Expand $(x + 10) (x + 10)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$x^{2} + x (x + 10) + 10 (x + 10) - {(x + 20)}^{2} = 0$

Step 2.1.2.2

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 10 + 10 (x + 10) - {(x + 20)}^{2} = 0$

Step 2.1.2.3

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 10 + 10 x + 10 \cdot 10 - {(x + 20)}^{2} = 0$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot 10 + 10 x + 10 \cdot 10 - {(x + 20)}^{2} = 0$

Step 2.1.3.1.2

Move $10$ to the left of $x$ .

$x^{2} + x^{2} + 10 \cdot x + 10 x + 10 \cdot 10 - {(x + 20)}^{2} = 0$

Step 2.1.3.1.3

Multiply $10$ by $10$ .

$x^{2} + x^{2} + 10 x + 10 x + 100 - {(x + 20)}^{2} = 0$

Step 2.1.3.2

Add $10 x$ and $10 x$ .

$x^{2} + x^{2} + 20 x + 100 - {(x + 20)}^{2} = 0$

Step 2.1.4

Rewrite ${(x + 20)}^{2}$ as $(x + 20) (x + 20)$ .

$x^{2} + x^{2} + 20 x + 100 - ((x + 20) (x + 20)) = 0$

Step 2.1.5

Expand $(x + 20) (x + 20)$ using the FOIL Method.

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Step 2.1.5.1

Apply the distributive property.

$x^{2} + x^{2} + 20 x + 100 - (x (x + 20) + 20 (x + 20)) = 0$

Step 2.1.5.2

Apply the distributive property.

$x^{2} + x^{2} + 20 x + 100 - (x \cdot x + x \cdot 20 + 20 (x + 20)) = 0$

Step 2.1.5.3

Apply the distributive property.

$x^{2} + x^{2} + 20 x + 100 - (x \cdot x + x \cdot 20 + 20 x + 20 \cdot 20) = 0$

Step 2.1.6

Simplify and combine like terms.

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Step 2.1.6.1

Simplify each term.

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Step 2.1.6.1.1

Multiply $x$ by $x$ .

$x^{2} + x^{2} + 20 x + 100 - (x^{2} + x \cdot 20 + 20 x + 20 \cdot 20) = 0$

Step 2.1.6.1.2

Move $20$ to the left of $x$ .

$x^{2} + x^{2} + 20 x + 100 - (x^{2} + 20 \cdot x + 20 x + 20 \cdot 20) = 0$

Step 2.1.6.1.3

Multiply $20$ by $20$ .

$x^{2} + x^{2} + 20 x + 100 - (x^{2} + 20 x + 20 x + 400) = 0$

Step 2.1.6.2

Add $20 x$ and $20 x$ .

$x^{2} + x^{2} + 20 x + 100 - (x^{2} + 40 x + 400) = 0$

Step 2.1.7

Apply the distributive property.

$x^{2} + x^{2} + 20 x + 100 - x^{2} - (40 x) - 1 \cdot 400 = 0$

Step 2.1.8

Simplify.

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Step 2.1.8.1

Multiply $40$ by $- 1$ .

$x^{2} + x^{2} + 20 x + 100 - x^{2} - 40 x - 1 \cdot 400 = 0$

Step 2.1.8.2

Multiply $- 1$ by $400$ .

$x^{2} + x^{2} + 20 x + 100 - x^{2} - 40 x - 400 = 0$

Step 2.2

Combine the opposite terms in $x^{2} + x^{2} + 20 x + 100 - x^{2} - 40 x - 400$ .

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Step 2.2.1

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

$x^{2} + 20 x + 100 + 0 - 40 x - 400 = 0$

Step 2.2.2

Add $x^{2} + 20 x + 100$ and $0$ .

$x^{2} + 20 x + 100 - 40 x - 400 = 0$

Step 2.3

Subtract $40 x$ from $20 x$ .

$x^{2} - 20 x + 100 - 400 = 0$

Step 2.4

Subtract $400$ from $100$ .

$x^{2} - 20 x - 300 = 0$

Step 3

Factor $x^{2} - 20 x - 300$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 300$ and whose sum is $- 20$ .

$- 30, 10$

Step 3.2

Write the factored form using these integers.

$(x - 30) (x + 10) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 30 = 0$

$x + 10 = 0$

Step 5

Set $x - 30$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x - 30$ equal to $0$ .

$x - 30 = 0$

Step 5.2

Add $30$ to both sides of the equation.

$x = 30$

Step 6

Set $x + 10$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x + 10$ equal to $0$ .

$x + 10 = 0$

Step 6.2

Subtract $10$ from both sides of the equation.

$x = - 10$

Step 7

The final solution is all the values that make $(x - 30) (x + 10) = 0$ true.

$x = 30, - 10$