Finite Math Examples

$(x - 3) (1 - x) = 1$

Step 1

Subtract $1$ from both sides of the equation.

$(x - 3) (1 - x) - 1 = 0$

Step 2

Simplify $(x - 3) (1 - x) - 1$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Expand $(x - 3) (1 - x)$ using the FOIL Method.

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Step 2.1.1.1

Apply the distributive property.

$x (1 - x) - 3 (1 - x) - 1 = 0$

Step 2.1.1.2

Apply the distributive property.

$x \cdot 1 + x (- x) - 3 (1 - x) - 1 = 0$

Step 2.1.1.3

Apply the distributive property.

$x \cdot 1 + x (- x) - 3 \cdot 1 - 3 (- x) - 1 = 0$

Step 2.1.2

Simplify and combine like terms.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

Multiply $x$ by $1$ .

$x + x (- x) - 3 \cdot 1 - 3 (- x) - 1 = 0$

Step 2.1.2.1.2

Rewrite using the commutative property of multiplication.

$x - x \cdot x - 3 \cdot 1 - 3 (- x) - 1 = 0$

Step 2.1.2.1.3

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

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Step 2.1.2.1.3.1

Move $x$ .

$x - (x \cdot x) - 3 \cdot 1 - 3 (- x) - 1 = 0$

Step 2.1.2.1.3.2

Multiply $x$ by $x$ .

$x - x^{2} - 3 \cdot 1 - 3 (- x) - 1 = 0$

Step 2.1.2.1.4

Multiply $- 3$ by $1$ .

$x - x^{2} - 3 - 3 (- x) - 1 = 0$

Step 2.1.2.1.5

Multiply $- 1$ by $- 3$ .

$x - x^{2} - 3 + 3 x - 1 = 0$

Step 2.1.2.2

Add $x$ and $3 x$ .

$4 x - x^{2} - 3 - 1 = 0$

Step 2.2

Subtract $1$ from $- 3$ .

$4 x - x^{2} - 4 = 0$

Step 3

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

$4 u - u^{2} - 4 = 0$

Step 4

Factor by grouping.

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Step 4.1

Reorder terms.

$- u^{2} + 4 u - 4 = 0$

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 $a \cdot c = - 1 \cdot - 4 = 4$ and whose sum is $b = 4$ .

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Step 4.2.1

Factor $4$ out of $4 u$ .

$- u^{2} + 4 (u) - 4 = 0$

Step 4.2.2

Rewrite $4$ as $2$ plus $2$

$- u^{2} + (2 + 2) u - 4 = 0$

Step 4.2.3

Apply the distributive property.

$- u^{2} + 2 u + 2 u - 4 = 0$

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.

$(- u^{2} + 2 u) + 2 u - 4 = 0$

Step 4.3.2

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

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

Step 4.4

Factor the polynomial by factoring out the greatest common factor, $- u + 2$ .

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

Step 5

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

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

Step 6

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

$- x + 2 = 0$

$x - 2 = 0$

Step 7

Set $- x + 2$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $- x + 2$ equal to $0$ .

$- x + 2 = 0$

Step 7.2

Solve $- x + 2 = 0$ for $x$ .

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Step 7.2.1

Subtract $2$ from both sides of the equation.

$- x = - 2$

Step 7.2.2

Divide each term in $- x = - 2$ by $- 1$ and simplify.

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Step 7.2.2.1

Divide each term in $- x = - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 2}{- 1}$

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.

$\frac{x}{1} = \frac{- 2}{- 1}$

Step 7.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 1}$

Step 7.2.2.3

Simplify the right side.

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Step 7.2.2.3.1

Divide $- 2$ by $- 1$ .

$x = 2$

Step 8

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

$x = 2$