Pre-Algebra Examples

$(x - 1) (x + 2) = 2 {(x - 1)}^{2}$

Step 1

Simplify $(x - 1) (x + 2)$ .

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Step 1.1

Rewrite.

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

Step 1.2

Simplify by adding zeros.

$(x - 1) (x + 2) = 2 {(x - 1)}^{2}$

Step 1.3

Expand $(x - 1) (x + 2)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$x (x + 2) - 1 (x + 2) = 2 {(x - 1)}^{2}$

Step 1.3.2

Apply the distributive property.

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

Step 1.3.3

Apply the distributive property.

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

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $x$ by $x$ .

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

Step 1.4.1.2

Move $2$ to the left of $x$ .

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

Step 1.4.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 1.4.1.4

Multiply $- 1$ by $2$ .

$x^{2} + 2 x - x - 2 = 2 {(x - 1)}^{2}$

Step 1.4.2

Subtract $x$ from $2 x$ .

$x^{2} + x - 2 = 2 {(x - 1)}^{2}$

Step 2

Simplify $2 {(x - 1)}^{2}$ .

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Step 2.1

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

$x^{2} + x - 2 = 2 ((x - 1) (x - 1))$

Step 2.2

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

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Step 2.2.1

Apply the distributive property.

$x^{2} + x - 2 = 2 (x (x - 1) - 1 (x - 1))$

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

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

Multiply $x$ by $x$ .

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

Step 2.3.1.2

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

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

Step 2.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 2.3.1.5

Multiply $- 1$ by $- 1$ .

$x^{2} + x - 2 = 2 (x^{2} - x - x + 1)$

Step 2.3.2

Subtract $x$ from $- x$ .

$x^{2} + x - 2 = 2 (x^{2} - 2 x + 1)$

Step 2.4

Apply the distributive property.

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

Step 2.5

Simplify.

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Step 2.5.1

Multiply $- 2$ by $2$ .

$x^{2} + x - 2 = 2 x^{2} - 4 x + 2 \cdot 1$

Step 2.5.2

Multiply $2$ by $1$ .

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Add $4 x$ to both sides of the equation.

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

Step 3.3

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

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

Step 3.4

Add $x$ and $4 x$ .

$- x^{2} + 5 x - 2 = 2$

Step 4

Subtract $2$ from both sides of the equation.

$- x^{2} + 5 x - 2 - 2 = 0$

Step 5

Subtract $2$ from $- 2$ .

$- x^{2} + 5 x - 4 = 0$

Step 6

Factor the left side of the equation.

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Step 6.1

Factor $- 1$ out of $- x^{2} + 5 x - 4$ .

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Step 6.1.1

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 5 x - 4 = 0$

Step 6.1.2

Factor $- 1$ out of $5 x$ .

$- (x^{2}) - (- 5 x) - 4 = 0$

Step 6.1.3

Rewrite $- 4$ as $- 1 (4)$ .

$- (x^{2}) - (- 5 x) - 1 \cdot 4 = 0$

Step 6.1.4

Factor $- 1$ out of $- (x^{2}) - (- 5 x)$ .

$- (x^{2} - 5 x) - 1 \cdot 4 = 0$

Step 6.1.5

Factor $- 1$ out of $- (x^{2} - 5 x) - 1 (4)$ .

$- (x^{2} - 5 x + 4) = 0$

Step 6.2

Factor.

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Step 6.2.1

Factor $x^{2} - 5 x + 4$ using the AC method.

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Step 6.2.1.1

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 $4$ and whose sum is $- 5$ .

$- 4, - 1$

Step 6.2.1.2

Write the factored form using these integers.

$- ((x - 4) (x - 1)) = 0$

Step 6.2.2

Remove unnecessary parentheses.

$- (x - 4) (x - 1) = 0$

Step 7

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

$x - 4 = 0$

$x - 1 = 0$

Step 8

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 8.2

Add $4$ to both sides of the equation.

$x = 4$

Step 9

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 9.2

Add $1$ to both sides of the equation.

$x = 1$

Step 10

The final solution is all the values that make $- (x - 4) (x - 1) = 0$ true.

$x = 4, 1$