Finite Math Examples

${(1 - y)}^{2} + (1 - y) y - y^{2} = - 41$

Step 1

Add $41$ to both sides of the equation.

${(1 - y)}^{2} + (1 - y) y - y^{2} + 41 = 0$

Step 2

Simplify ${(1 - y)}^{2} + (1 - y) y - y^{2} + 41$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$(1 - y) (1 - y) + (1 - y) y - y^{2} + 41 = 0$

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.

$1 (1 - y) - y (1 - y) + (1 - y) y - y^{2} + 41 = 0$

Step 2.1.2.2

Apply the distributive property.

$1 \cdot 1 + 1 (- y) - y (1 - y) + (1 - y) y - y^{2} + 41 = 0$

Step 2.1.2.3

Apply the distributive property.

$1 \cdot 1 + 1 (- y) - y \cdot 1 - y (- y) + (1 - y) y - y^{2} + 41 = 0$

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 $1$ .

$1 + 1 (- y) - y \cdot 1 - y (- y) + (1 - y) y - y^{2} + 41 = 0$

Step 2.1.3.1.2

Multiply $- y$ by $1$ .

$1 - y - y \cdot 1 - y (- y) + (1 - y) y - y^{2} + 41 = 0$

Step 2.1.3.1.3

Multiply $- 1$ by $1$ .

$1 - y - y - y (- y) + (1 - y) y - y^{2} + 41 = 0$

Step 2.1.3.1.4

Rewrite using the commutative property of multiplication.

$1 - y - y - 1 \cdot - 1 y \cdot y + (1 - y) y - y^{2} + 41 = 0$

Step 2.1.3.1.5

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

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Step 2.1.3.1.5.1

Move $y$ .

$1 - y - y - 1 \cdot - 1 (y \cdot y) + (1 - y) y - y^{2} + 41 = 0$

Step 2.1.3.1.5.2

Multiply $y$ by $y$ .

$1 - y - y - 1 \cdot - 1 y^{2} + (1 - y) y - y^{2} + 41 = 0$

Step 2.1.3.1.6

Multiply $- 1$ by $- 1$ .

$1 - y - y + 1 y^{2} + (1 - y) y - y^{2} + 41 = 0$

Step 2.1.3.1.7

Multiply $y^{2}$ by $1$ .

$1 - y - y + y^{2} + (1 - y) y - y^{2} + 41 = 0$

Step 2.1.3.2

Subtract $y$ from $- y$ .

$1 - 2 y + y^{2} + (1 - y) y - y^{2} + 41 = 0$

Step 2.1.4

Apply the distributive property.

$1 - 2 y + y^{2} + 1 y - y \cdot y - y^{2} + 41 = 0$

Step 2.1.5

Multiply $y$ by $1$ .

$1 - 2 y + y^{2} + y - y \cdot y - y^{2} + 41 = 0$

Step 2.1.6

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

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Step 2.1.6.1

Move $y$ .

$1 - 2 y + y^{2} + y - (y \cdot y) - y^{2} + 41 = 0$

Step 2.1.6.2

Multiply $y$ by $y$ .

$1 - 2 y + y^{2} + y - y^{2} - y^{2} + 41 = 0$

Step 2.2

Combine the opposite terms in $1 - 2 y + y^{2} + y - y^{2} - y^{2} + 41$ .

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Step 2.2.1

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

$1 - 2 y + y + 0 - y^{2} + 41 = 0$

Step 2.2.2

Add $1 - 2 y + y$ and $0$ .

$1 - 2 y + y - y^{2} + 41 = 0$

Step 2.3

Add $1$ and $41$ .

$- 2 y + y - y^{2} + 42 = 0$

Step 2.4

Add $- 2 y$ and $y$ .

$- y - y^{2} + 42 = 0$

Step 3

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

$- u - u^{2} + 42 = 0$

Step 4

Factor by grouping.

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Step 4.1

Reorder terms.

$- u^{2} - u + 42 = 0$

Step 4.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 42 = - 42$ and whose sum is $b = - 1$ .

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Step 4.2.1

Factor $- 1$ out of $- u$ .

$- u^{2} - (u) + 42 = 0$

Step 4.2.2

Rewrite $- 1$ as $6$ plus $- 7$

$- u^{2} + (6 - 7) u + 42 = 0$

Step 4.2.3

Apply the distributive property.

$- u^{2} + 6 u - 7 u + 42 = 0$

Step 4.3

Factor out the greatest common factor from each group.

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Step 4.3.1

Group the first two terms and the last two terms.

$(- u^{2} + 6 u) - 7 u + 42 = 0$

Step 4.3.2

Factor out the greatest common factor (GCF) from each group.

$u (- u + 6) + 7 (- u + 6) = 0$

Step 4.4

Factor the polynomial by factoring out the greatest common factor, $- u + 6$ .

$(- u + 6) (u + 7) = 0$

Step 5

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

$(- y + 6) (y + 7) = 0$

Step 6

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

$- y + 6 = 0$

$y + 7 = 0$

Step 7

Set $- y + 6$ equal to $0$ and solve for $y$ .

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Step 7.1

Set $- y + 6$ equal to $0$ .

$- y + 6 = 0$

Step 7.2

Solve $- y + 6 = 0$ for $y$ .

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Step 7.2.1

Subtract $6$ from both sides of the equation.

$- y = - 6$

Step 7.2.2

Divide each term in $- y = - 6$ by $- 1$ and simplify.

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Step 7.2.2.1

Divide each term in $- y = - 6$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 6}{- 1}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 6}{- 1}$

Step 7.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 6}{- 1}$

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Divide $- 6$ by $- 1$ .

$y = 6$

Step 8

Set $y + 7$ equal to $0$ and solve for $y$ .

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Step 8.1

Set $y + 7$ equal to $0$ .

$y + 7 = 0$

Step 8.2

Subtract $7$ from both sides of the equation.

$y = - 7$

Step 9

The final solution is all the values that make $(- y + 6) (y + 7) = 0$ true.

$y = 6, - 7$