Pre-Algebra Examples

$X (X - 1) = 2 (X - 1) (X + 5) (X - 4) (X - 4)$

Step 1

Simplify $X (X - 1)$ .

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Step 1.1

Simplify by multiplying through.

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Step 1.1.1

Apply the distributive property.

$X \cdot X + X \cdot - 1 = 2 (X - 1) (X + 5) (X - 4) (X - 4)$

Step 1.1.2

Simplify the expression.

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Step 1.1.2.1

Multiply $X$ by $X$ .

$X^{2} + X \cdot - 1 = 2 (X - 1) (X + 5) (X - 4) (X - 4)$

Step 1.1.2.2

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

$X^{2} - 1 \cdot X = 2 (X - 1) (X + 5) (X - 4) (X - 4)$

Step 1.2

Rewrite $- 1 X$ as $- X$ .

$X^{2} - X = 2 (X - 1) (X + 5) (X - 4) (X - 4)$

Step 2

Simplify $2 (X - 1) (X + 5) (X - 4) (X - 4)$ .

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Step 2.1

Simplify by multiplying through.

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Step 2.1.1

Apply the distributive property.

$X^{2} - X = (2 X + 2 \cdot - 1) (X + 5) (X - 4) (X - 4)$

Step 2.1.2

Multiply $2$ by $- 1$ .

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

Step 2.2

Expand $(2 X - 2) (X + 5)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

$X^{2} - X = (2 X \cdot X + 2 X \cdot 5 - 2 (X + 5)) (X - 4) (X - 4)$

Step 2.2.3

Apply the distributive property.

$X^{2} - X = (2 X \cdot X + 2 X \cdot 5 - 2 X - 2 \cdot 5) (X - 4) (X - 4)$

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 $X$ by adding the exponents.

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Step 2.3.1.1.1

Move $X$ .

$X^{2} - X = (2 (X \cdot X) + 2 X \cdot 5 - 2 X - 2 \cdot 5) (X - 4) (X - 4)$

Step 2.3.1.1.2

Multiply $X$ by $X$ .

$X^{2} - X = (2 X^{2} + 2 X \cdot 5 - 2 X - 2 \cdot 5) (X - 4) (X - 4)$

Step 2.3.1.2

Multiply $5$ by $2$ .

$X^{2} - X = (2 X^{2} + 10 X - 2 X - 2 \cdot 5) (X - 4) (X - 4)$

Step 2.3.1.3

Multiply $- 2$ by $5$ .

$X^{2} - X = (2 X^{2} + 10 X - 2 X - 10) (X - 4) (X - 4)$

Step 2.3.2

Subtract $2 X$ from $10 X$ .

$X^{2} - X = (2 X^{2} + 8 X - 10) (X - 4) (X - 4)$

Step 2.4

Expand $(2 X^{2} + 8 X - 10) (X - 4)$ by multiplying each term in the first expression by each term in the second expression.

$X^{2} - X = (2 X^{2} X + 2 X^{2} \cdot - 4 + 8 X \cdot X + 8 X \cdot - 4 - 10 X - 10 \cdot - 4) (X - 4)$

Step 2.5

Simplify terms.

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Step 2.5.1

Simplify each term.

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Step 2.5.1.1

Multiply $X^{2}$ by $X$ by adding the exponents.

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Step 2.5.1.1.1

Move $X$ .

$X^{2} - X = (2 (X \cdot X^{2}) + 2 X^{2} \cdot - 4 + 8 X \cdot X + 8 X \cdot - 4 - 10 X - 10 \cdot - 4) (X - 4)$

Step 2.5.1.1.2

Multiply $X$ by $X^{2}$ .

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Step 2.5.1.1.2.1

Raise $X$ to the power of $1$ .

$X^{2} - X = (2 (X^{1} X^{2}) + 2 X^{2} \cdot - 4 + 8 X \cdot X + 8 X \cdot - 4 - 10 X - 10 \cdot - 4) (X - 4)$

Step 2.5.1.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$X^{2} - X = (2 X^{1 + 2} + 2 X^{2} \cdot - 4 + 8 X \cdot X + 8 X \cdot - 4 - 10 X - 10 \cdot - 4) (X - 4)$

Step 2.5.1.1.3

Add $1$ and $2$ .

$X^{2} - X = (2 X^{3} + 2 X^{2} \cdot - 4 + 8 X \cdot X + 8 X \cdot - 4 - 10 X - 10 \cdot - 4) (X - 4)$

Step 2.5.1.2

Multiply $- 4$ by $2$ .

$X^{2} - X = (2 X^{3} - 8 X^{2} + 8 X \cdot X + 8 X \cdot - 4 - 10 X - 10 \cdot - 4) (X - 4)$

Step 2.5.1.3

Multiply $X$ by $X$ by adding the exponents.

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Step 2.5.1.3.1

Move $X$ .

$X^{2} - X = (2 X^{3} - 8 X^{2} + 8 (X \cdot X) + 8 X \cdot - 4 - 10 X - 10 \cdot - 4) (X - 4)$

Step 2.5.1.3.2

Multiply $X$ by $X$ .

$X^{2} - X = (2 X^{3} - 8 X^{2} + 8 X^{2} + 8 X \cdot - 4 - 10 X - 10 \cdot - 4) (X - 4)$

Step 2.5.1.4

Multiply $- 4$ by $8$ .

$X^{2} - X = (2 X^{3} - 8 X^{2} + 8 X^{2} - 32 X - 10 X - 10 \cdot - 4) (X - 4)$

Step 2.5.1.5

Multiply $- 10$ by $- 4$ .

$X^{2} - X = (2 X^{3} - 8 X^{2} + 8 X^{2} - 32 X - 10 X + 40) (X - 4)$

Step 2.5.2

Simplify by adding terms.

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Step 2.5.2.1

Combine the opposite terms in $2 X^{3} - 8 X^{2} + 8 X^{2} - 32 X - 10 X + 40$ .

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Step 2.5.2.1.1

Add $- 8 X^{2}$ and $8 X^{2}$ .

$X^{2} - X = (2 X^{3} + 0 - 32 X - 10 X + 40) (X - 4)$

Step 2.5.2.1.2

Add $2 X^{3}$ and $0$ .

$X^{2} - X = (2 X^{3} - 32 X - 10 X + 40) (X - 4)$

Step 2.5.2.2

Subtract $10 X$ from $- 32 X$ .

$X^{2} - X = (2 X^{3} - 42 X + 40) (X - 4)$

Step 2.6

Expand $(2 X^{3} - 42 X + 40) (X - 4)$ by multiplying each term in the first expression by each term in the second expression.

$X^{2} - X = 2 X^{3} X + 2 X^{3} \cdot - 4 - 42 X \cdot X - 42 X \cdot - 4 + 40 X + 40 \cdot - 4$

Step 2.7

Simplify terms.

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Step 2.7.1

Simplify each term.

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Step 2.7.1.1

Multiply $X^{3}$ by $X$ by adding the exponents.

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Step 2.7.1.1.1

Move $X$ .

$X^{2} - X = 2 (X \cdot X^{3}) + 2 X^{3} \cdot - 4 - 42 X \cdot X - 42 X \cdot - 4 + 40 X + 40 \cdot - 4$

Step 2.7.1.1.2

Multiply $X$ by $X^{3}$ .

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Step 2.7.1.1.2.1

Raise $X$ to the power of $1$ .

$X^{2} - X = 2 (X^{1} X^{3}) + 2 X^{3} \cdot - 4 - 42 X \cdot X - 42 X \cdot - 4 + 40 X + 40 \cdot - 4$

Step 2.7.1.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$X^{2} - X = 2 X^{1 + 3} + 2 X^{3} \cdot - 4 - 42 X \cdot X - 42 X \cdot - 4 + 40 X + 40 \cdot - 4$

Step 2.7.1.1.3

Add $1$ and $3$ .

$X^{2} - X = 2 X^{4} + 2 X^{3} \cdot - 4 - 42 X \cdot X - 42 X \cdot - 4 + 40 X + 40 \cdot - 4$

Step 2.7.1.2

Multiply $- 4$ by $2$ .

$X^{2} - X = 2 X^{4} - 8 X^{3} - 42 X \cdot X - 42 X \cdot - 4 + 40 X + 40 \cdot - 4$

Step 2.7.1.3

Multiply $X$ by $X$ by adding the exponents.

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Step 2.7.1.3.1

Move $X$ .

$X^{2} - X = 2 X^{4} - 8 X^{3} - 42 (X \cdot X) - 42 X \cdot - 4 + 40 X + 40 \cdot - 4$

Step 2.7.1.3.2

Multiply $X$ by $X$ .

$X^{2} - X = 2 X^{4} - 8 X^{3} - 42 X^{2} - 42 X \cdot - 4 + 40 X + 40 \cdot - 4$

Step 2.7.1.4

Multiply $- 4$ by $- 42$ .

$X^{2} - X = 2 X^{4} - 8 X^{3} - 42 X^{2} + 168 X + 40 X + 40 \cdot - 4$

Step 2.7.1.5

Multiply $40$ by $- 4$ .

$X^{2} - X = 2 X^{4} - 8 X^{3} - 42 X^{2} + 168 X + 40 X - 160$

Step 2.7.2

Add $168 X$ and $40 X$ .

$X^{2} - X = 2 X^{4} - 8 X^{3} - 42 X^{2} + 208 X - 160$

Step 3

Graph each side of the equation. The solution is the x-value of the point of intersection.

$X \approx - 5.03084282, 1, 3.54456995, 4.48627286$

Step 4