Basic Math Examples

${(2 y - 120)}^{2} = y^{2} + {(y + 120)}^{2}$

Step 1

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

$y^{2} + {(y + 120)}^{2} = {(2 y - 120)}^{2}$

Step 2

Simplify ${(2 y - 120)}^{2}$ .

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Step 2.1

Rewrite ${(2 y - 120)}^{2}$ as $(2 y - 120) (2 y - 120)$ .

$y^{2} + {(y + 120)}^{2} = (2 y - 120) (2 y - 120)$

Step 2.2

Expand $(2 y - 120) (2 y - 120)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$y^{2} + {(y + 120)}^{2} = 2 y (2 y - 120) - 120 (2 y - 120)$

Step 2.2.2

Apply the distributive property.

$y^{2} + {(y + 120)}^{2} = 2 y (2 y) + 2 y \cdot - 120 - 120 (2 y - 120)$

Step 2.2.3

Apply the distributive property.

$y^{2} + {(y + 120)}^{2} = 2 y (2 y) + 2 y \cdot - 120 - 120 (2 y) - 120 \cdot - 120$

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

Rewrite using the commutative property of multiplication.

$y^{2} + {(y + 120)}^{2} = 2 \cdot 2 y \cdot y + 2 y \cdot - 120 - 120 (2 y) - 120 \cdot - 120$

Step 2.3.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 2.3.1.2.1

Move $y$ .

$y^{2} + {(y + 120)}^{2} = 2 \cdot 2 (y \cdot y) + 2 y \cdot - 120 - 120 (2 y) - 120 \cdot - 120$

Step 2.3.1.2.2

Multiply $y$ by $y$ .

$y^{2} + {(y + 120)}^{2} = 2 \cdot 2 y^{2} + 2 y \cdot - 120 - 120 (2 y) - 120 \cdot - 120$

Step 2.3.1.3

Multiply $2$ by $2$ .

$y^{2} + {(y + 120)}^{2} = 4 y^{2} + 2 y \cdot - 120 - 120 (2 y) - 120 \cdot - 120$

Step 2.3.1.4

Multiply $- 120$ by $2$ .

$y^{2} + {(y + 120)}^{2} = 4 y^{2} - 240 y - 120 (2 y) - 120 \cdot - 120$

Step 2.3.1.5

Multiply $2$ by $- 120$ .

$y^{2} + {(y + 120)}^{2} = 4 y^{2} - 240 y - 240 y - 120 \cdot - 120$

Step 2.3.1.6

Multiply $- 120$ by $- 120$ .

$y^{2} + {(y + 120)}^{2} = 4 y^{2} - 240 y - 240 y + 14400$

Step 2.3.2

Subtract $240 y$ from $- 240 y$ .

$y^{2} + {(y + 120)}^{2} = 4 y^{2} - 480 y + 14400$

Step 3

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

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Step 3.1

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

$y^{2} + {(y + 120)}^{2} - 4 y^{2} = - 480 y + 14400$

Step 3.2

Add $480 y$ to both sides of the equation.

$y^{2} + {(y + 120)}^{2} - 4 y^{2} + 480 y = 14400$

Step 3.3

Simplify each term.

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Step 3.3.1

Rewrite ${(y + 120)}^{2}$ as $(y + 120) (y + 120)$ .

$y^{2} + (y + 120) (y + 120) - 4 y^{2} + 480 y = 14400$

Step 3.3.2

Expand $(y + 120) (y + 120)$ using the FOIL Method.

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Step 3.3.2.1

Apply the distributive property.

$y^{2} + y (y + 120) + 120 (y + 120) - 4 y^{2} + 480 y = 14400$

Step 3.3.2.2

Apply the distributive property.

$y^{2} + y \cdot y + y \cdot 120 + 120 (y + 120) - 4 y^{2} + 480 y = 14400$

Step 3.3.2.3

Apply the distributive property.

$y^{2} + y \cdot y + y \cdot 120 + 120 y + 120 \cdot 120 - 4 y^{2} + 480 y = 14400$

Step 3.3.3

Simplify and combine like terms.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Multiply $y$ by $y$ .

$y^{2} + y^{2} + y \cdot 120 + 120 y + 120 \cdot 120 - 4 y^{2} + 480 y = 14400$

Step 3.3.3.1.2

Move $120$ to the left of $y$ .

$y^{2} + y^{2} + 120 \cdot y + 120 y + 120 \cdot 120 - 4 y^{2} + 480 y = 14400$

Step 3.3.3.1.3

Multiply $120$ by $120$ .

$y^{2} + y^{2} + 120 y + 120 y + 14400 - 4 y^{2} + 480 y = 14400$

Step 3.3.3.2

Add $120 y$ and $120 y$ .

$y^{2} + y^{2} + 240 y + 14400 - 4 y^{2} + 480 y = 14400$

Step 3.4

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

$2 y^{2} + 240 y + 14400 - 4 y^{2} + 480 y = 14400$

Step 3.5

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

$- 2 y^{2} + 240 y + 14400 + 480 y = 14400$

Step 3.6

Add $240 y$ and $480 y$ .

$- 2 y^{2} + 720 y + 14400 = 14400$

Step 4

Subtract $14400$ from both sides of the equation.

$- 2 y^{2} + 720 y + 14400 - 14400 = 0$

Step 5

Combine the opposite terms in $- 2 y^{2} + 720 y + 14400 - 14400$ .

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Step 5.1

Subtract $14400$ from $14400$ .

$- 2 y^{2} + 720 y + 0 = 0$

Step 5.2

Add $- 2 y^{2} + 720 y$ and $0$ .

$- 2 y^{2} + 720 y = 0$

Step 6

Factor $- 2 y$ out of $- 2 y^{2} + 720 y$ .

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Step 6.1

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

$- 2 y \cdot y + 720 y = 0$

Step 6.2

Factor $- 2 y$ out of $720 y$ .

$- 2 y \cdot y - 2 y \cdot - 360 = 0$

Step 6.3

Factor $- 2 y$ out of $- 2 y (y) - 2 y (- 360)$ .

$- 2 y (y - 360) = 0$

Step 7

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

$y = 0$

$y - 360 = 0$

Step 8

Set $y$ equal to $0$ .

$y = 0$

Step 9

Set $y - 360$ equal to $0$ and solve for $y$ .

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Step 9.1

Set $y - 360$ equal to $0$ .

$y - 360 = 0$

Step 9.2

Add $360$ to both sides of the equation.

$y = 360$

Step 10

The final solution is all the values that make $- 2 y (y - 360) = 0$ true.

$y = 0, 360$