Pre-Algebra Examples

$x^{2} + {(x + 15)}^{2} = {(2 x - 15)}^{2}$

Step 1

Simplify ${(2 x - 15)}^{2}$ .

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Step 1.1

Rewrite ${(2 x - 15)}^{2}$ as $(2 x - 15) (2 x - 15)$ .

$x^{2} + {(x + 15)}^{2} = (2 x - 15) (2 x - 15)$

Step 1.2

Expand $(2 x - 15) (2 x - 15)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$x^{2} + {(x + 15)}^{2} = 2 x (2 x - 15) - 15 (2 x - 15)$

Step 1.2.2

Apply the distributive property.

$x^{2} + {(x + 15)}^{2} = 2 x (2 x) + 2 x \cdot - 15 - 15 (2 x - 15)$

Step 1.2.3

Apply the distributive property.

$x^{2} + {(x + 15)}^{2} = 2 x (2 x) + 2 x \cdot - 15 - 15 (2 x) - 15 \cdot - 15$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$x^{2} + {(x + 15)}^{2} = 2 \cdot 2 x \cdot x + 2 x \cdot - 15 - 15 (2 x) - 15 \cdot - 15$

Step 1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.1.2.1

Move $x$ .

$x^{2} + {(x + 15)}^{2} = 2 \cdot 2 (x \cdot x) + 2 x \cdot - 15 - 15 (2 x) - 15 \cdot - 15$

Step 1.3.1.2.2

Multiply $x$ by $x$ .

$x^{2} + {(x + 15)}^{2} = 2 \cdot 2 x^{2} + 2 x \cdot - 15 - 15 (2 x) - 15 \cdot - 15$

Step 1.3.1.3

Multiply $2$ by $2$ .

$x^{2} + {(x + 15)}^{2} = 4 x^{2} + 2 x \cdot - 15 - 15 (2 x) - 15 \cdot - 15$

Step 1.3.1.4

Multiply $- 15$ by $2$ .

$x^{2} + {(x + 15)}^{2} = 4 x^{2} - 30 x - 15 (2 x) - 15 \cdot - 15$

Step 1.3.1.5

Multiply $2$ by $- 15$ .

$x^{2} + {(x + 15)}^{2} = 4 x^{2} - 30 x - 30 x - 15 \cdot - 15$

Step 1.3.1.6

Multiply $- 15$ by $- 15$ .

$x^{2} + {(x + 15)}^{2} = 4 x^{2} - 30 x - 30 x + 225$

Step 1.3.2

Subtract $30 x$ from $- 30 x$ .

$x^{2} + {(x + 15)}^{2} = 4 x^{2} - 60 x + 225$

Step 2

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

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Step 2.1

Subtract $4 x^{2}$ from both sides of the equation.

$x^{2} + {(x + 15)}^{2} - 4 x^{2} = - 60 x + 225$

Step 2.2

Add $60 x$ to both sides of the equation.

$x^{2} + {(x + 15)}^{2} - 4 x^{2} + 60 x = 225$

Step 2.3

Simplify each term.

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Step 2.3.1

Rewrite ${(x + 15)}^{2}$ as $(x + 15) (x + 15)$ .

$x^{2} + (x + 15) (x + 15) - 4 x^{2} + 60 x = 225$

Step 2.3.2

Expand $(x + 15) (x + 15)$ using the FOIL Method.

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Step 2.3.2.1

Apply the distributive property.

$x^{2} + x (x + 15) + 15 (x + 15) - 4 x^{2} + 60 x = 225$

Step 2.3.2.2

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 15 + 15 (x + 15) - 4 x^{2} + 60 x = 225$

Step 2.3.2.3

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 15 + 15 x + 15 \cdot 15 - 4 x^{2} + 60 x = 225$

Step 2.3.3

Simplify and combine like terms.

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Step 2.3.3.1

Simplify each term.

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Step 2.3.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot 15 + 15 x + 15 \cdot 15 - 4 x^{2} + 60 x = 225$

Step 2.3.3.1.2

Move $15$ to the left of $x$ .

$x^{2} + x^{2} + 15 \cdot x + 15 x + 15 \cdot 15 - 4 x^{2} + 60 x = 225$

Step 2.3.3.1.3

Multiply $15$ by $15$ .

$x^{2} + x^{2} + 15 x + 15 x + 225 - 4 x^{2} + 60 x = 225$

Step 2.3.3.2

Add $15 x$ and $15 x$ .

$x^{2} + x^{2} + 30 x + 225 - 4 x^{2} + 60 x = 225$

Step 2.4

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} + 30 x + 225 - 4 x^{2} + 60 x = 225$

Step 2.5

Subtract $4 x^{2}$ from $2 x^{2}$ .

$- 2 x^{2} + 30 x + 225 + 60 x = 225$

Step 2.6

Add $30 x$ and $60 x$ .

$- 2 x^{2} + 90 x + 225 = 225$

Step 3

Subtract $225$ from both sides of the equation.

$- 2 x^{2} + 90 x + 225 - 225 = 0$

Step 4

Combine the opposite terms in $- 2 x^{2} + 90 x + 225 - 225$ .

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Step 4.1

Subtract $225$ from $225$ .

$- 2 x^{2} + 90 x + 0 = 0$

Step 4.2

Add $- 2 x^{2} + 90 x$ and $0$ .

$- 2 x^{2} + 90 x = 0$

Step 5

Factor $- 2 x$ out of $- 2 x^{2} + 90 x$ .

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Step 5.1

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

$- 2 x \cdot x + 90 x = 0$

Step 5.2

Factor $- 2 x$ out of $90 x$ .

$- 2 x \cdot x - 2 x \cdot - 45 = 0$

Step 5.3

Factor $- 2 x$ out of $- 2 x (x) - 2 x (- 45)$ .

$- 2 x (x - 45) = 0$

Step 6

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

$x = 0$

$x - 45 = 0$

Step 7

Set $x$ equal to $0$ .

$x = 0$

Step 8

Set $x - 45$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x - 45$ equal to $0$ .

$x - 45 = 0$

Step 8.2

Add $45$ to both sides of the equation.

$x = 45$

Step 9

The final solution is all the values that make $- 2 x (x - 45) = 0$ true.

$x = 0, 45$