Finite Math Examples

${(- 1 + y)}^{2} + y^{2} = 421$

Step 1

Subtract $421$ from both sides of the equation.

${(- 1 + y)}^{2} + y^{2} - 421 = 0$

Step 2

Simplify ${(- 1 + y)}^{2} + y^{2} - 421$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite ${(- 1 + y)}^{2}$ as $(- 1 + y) (- 1 + y)$ .

$(- 1 + y) (- 1 + y) + y^{2} - 421 = 0$

Step 2.1.2

Expand $(- 1 + y) (- 1 + y)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

$- 1 (- 1 + y) + y (- 1 + y) + y^{2} - 421 = 0$

Step 2.1.2.2

Apply the distributive property.

$- 1 \cdot - 1 - 1 y + y (- 1 + y) + y^{2} - 421 = 0$

Step 2.1.2.3

Apply the distributive property.

$- 1 \cdot - 1 - 1 y + y \cdot - 1 + y \cdot y + y^{2} - 421 = 0$

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 $- 1$ by $- 1$ .

$1 - 1 y + y \cdot - 1 + y \cdot y + y^{2} - 421 = 0$

Step 2.1.3.1.2

Rewrite $- 1 y$ as $- y$ .

$1 - y + y \cdot - 1 + y \cdot y + y^{2} - 421 = 0$

Step 2.1.3.1.3

Move $- 1$ to the left of $y$ .

$1 - y - 1 \cdot y + y \cdot y + y^{2} - 421 = 0$

Step 2.1.3.1.4

Rewrite $- 1 y$ as $- y$ .

$1 - y - y + y \cdot y + y^{2} - 421 = 0$

Step 2.1.3.1.5

Multiply $y$ by $y$ .

$1 - y - y + y^{2} + y^{2} - 421 = 0$

Step 2.1.3.2

Subtract $y$ from $- y$ .

$1 - 2 y + y^{2} + y^{2} - 421 = 0$

Step 2.2

Subtract $421$ from $1$ .

$- 2 y + y^{2} + y^{2} - 420 = 0$

Step 2.3

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

$- 2 y + 2 y^{2} - 420 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $2$ out of $- 2 y + 2 y^{2} - 420$ .

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Step 3.1.1

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

$2 (- y) + 2 y^{2} - 420 = 0$

Step 3.1.2

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

$2 (- y) + 2 (y^{2}) - 420 = 0$

Step 3.1.3

Factor $2$ out of $- 420$ .

$2 (- y) + 2 y^{2} + 2 \cdot - 210 = 0$

Step 3.1.4

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

$2 (- y + y^{2}) + 2 \cdot - 210 = 0$

Step 3.1.5

Factor $2$ out of $2 (- y + y^{2}) + 2 \cdot - 210$ .

$2 (- y + y^{2} - 210) = 0$

Step 3.2

Let $u = y$ . Substitute $u$ for all occurrences of $y$ .

$2 (- u + u^{2} - 210) = 0$

Step 3.3

Factor $- u + u^{2} - 210$ using the AC method.

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Step 3.3.1

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 $- 210$ and whose sum is $- 1$ .

$- 15, 14$

Step 3.3.2

Write the factored form using these integers.

$2 ((u - 15) (u + 14)) = 0$

Step 3.4

Factor.

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Step 3.4.1

Replace all occurrences of $u$ with $y$ .

$2 ((y - 15) (y + 14)) = 0$

Step 3.4.2

Remove unnecessary parentheses.

$2 (y - 15) (y + 14) = 0$

Step 4

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

$y - 15 = 0$

$y + 14 = 0$

Step 5

Set $y - 15$ equal to $0$ and solve for $y$ .

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Step 5.1

Set $y - 15$ equal to $0$ .

$y - 15 = 0$

Step 5.2

Add $15$ to both sides of the equation.

$y = 15$

Step 6

Set $y + 14$ equal to $0$ and solve for $y$ .

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Step 6.1

Set $y + 14$ equal to $0$ .

$y + 14 = 0$

Step 6.2

Subtract $14$ from both sides of the equation.

$y = - 14$

Step 7

The final solution is all the values that make $2 (y - 15) (y + 14) = 0$ true.

$y = 15, - 14$