Pre-Algebra Examples

$\frac{(x^{2} - 1) \cdot ((2 x - 1) + \sqrt{3 x - 2})}{x - 1}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

$\frac{(x^{2} - 1^{2}) (2 x - 1 + \sqrt{3 x - 2})}{x - 1}$

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{(x + 1) (x - 1) (2 x - 1 + \sqrt{3 x - 2})}{x - 1}$

Step 2

Cancel the common factor of $x - 1$ .

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Step 2.1

Cancel the common factor.

$\frac{(x + 1) (x - 1) (2 x - 1 + \sqrt{3 x - 2})}{x - 1}$

Step 2.2

Divide $(x + 1) (2 x - 1 + \sqrt{3 x - 2})$ by $1$ .

$(x + 1) (2 x - 1 + \sqrt{3 x - 2})$

Step 3

Expand $(x + 1) (2 x - 1 + \sqrt{3 x - 2})$ by multiplying each term in the first expression by each term in the second expression.

$x (2 x) + x \cdot - 1 + x \sqrt{3 x - 2} + 1 (2 x) + 1 \cdot - 1 + 1 \sqrt{3 x - 2}$

Step 4

Simplify each term.

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Step 4.1

Rewrite using the commutative property of multiplication.

$2 x \cdot x + x \cdot - 1 + x \sqrt{3 x - 2} + 1 (2 x) + 1 \cdot - 1 + 1 \sqrt{3 x - 2}$

Step 4.2

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

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Step 4.2.1

Move $x$ .

$2 (x \cdot x) + x \cdot - 1 + x \sqrt{3 x - 2} + 1 (2 x) + 1 \cdot - 1 + 1 \sqrt{3 x - 2}$

Step 4.2.2

Multiply $x$ by $x$ .

$2 x^{2} + x \cdot - 1 + x \sqrt{3 x - 2} + 1 (2 x) + 1 \cdot - 1 + 1 \sqrt{3 x - 2}$

Step 4.3

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

$2 x^{2} - 1 \cdot x + x \sqrt{3 x - 2} + 1 (2 x) + 1 \cdot - 1 + 1 \sqrt{3 x - 2}$

Step 4.4

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

$2 x^{2} - x + x \sqrt{3 x - 2} + 1 (2 x) + 1 \cdot - 1 + 1 \sqrt{3 x - 2}$

Step 4.5

Multiply $2 x$ by $1$ .

$2 x^{2} - x + x \sqrt{3 x - 2} + 2 x + 1 \cdot - 1 + 1 \sqrt{3 x - 2}$

Step 4.6

Multiply $- 1$ by $1$ .

$2 x^{2} - x + x \sqrt{3 x - 2} + 2 x - 1 + 1 \sqrt{3 x - 2}$

Step 4.7

Multiply $\sqrt{3 x - 2}$ by $1$ .

$2 x^{2} - x + x \sqrt{3 x - 2} + 2 x - 1 + \sqrt{3 x - 2}$

Step 5

Add $- x$ and $2 x$ .

$2 x^{2} + x \sqrt{3 x - 2} + x - 1 + \sqrt{3 x - 2}$