Algebra Examples

$\frac{x (x - 1)}{2} - \frac{(x - 4) (x - 6)}{2} = 33$

Step 1

Simplify $\frac{x (x - 1)}{2} - \frac{(x - 4) (x - 6)}{2}$ .

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Step 1.1

Combine the numerators over the common denominator.

$\frac{x (x - 1) - (x - 4) (x - 6)}{2} = 33$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 1 - (x - 4) (x - 6)}{2} = 33$

Step 1.2.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 1 - (x - 4) (x - 6)}{2} = 33$

Step 1.2.3

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

$\frac{x^{2} - 1 \cdot x - (x - 4) (x - 6)}{2} = 33$

Step 1.2.4

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

$\frac{x^{2} - x - (x - 4) (x - 6)}{2} = 33$

Step 1.2.5

Apply the distributive property.

$\frac{x^{2} - x + (- x - - 4) (x - 6)}{2} = 33$

Step 1.2.6

Multiply $- 1$ by $- 4$ .

$\frac{x^{2} - x + (- x + 4) (x - 6)}{2} = 33$

Step 1.2.7

Expand $(- x + 4) (x - 6)$ using the FOIL Method.

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Step 1.2.7.1

Apply the distributive property.

$\frac{x^{2} - x - x (x - 6) + 4 (x - 6)}{2} = 33$

Step 1.2.7.2

Apply the distributive property.

$\frac{x^{2} - x - x \cdot x - x \cdot - 6 + 4 (x - 6)}{2} = 33$

Step 1.2.7.3

Apply the distributive property.

$\frac{x^{2} - x - x \cdot x - x \cdot - 6 + 4 x + 4 \cdot - 6}{2} = 33$

Step 1.2.8

Simplify and combine like terms.

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Step 1.2.8.1

Simplify each term.

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Step 1.2.8.1.1

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

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Step 1.2.8.1.1.1

Move $x$ .

$\frac{x^{2} - x - (x \cdot x) - x \cdot - 6 + 4 x + 4 \cdot - 6}{2} = 33$

Step 1.2.8.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} - x - x^{2} - x \cdot - 6 + 4 x + 4 \cdot - 6}{2} = 33$

Step 1.2.8.1.2

Multiply $- 6$ by $- 1$ .

$\frac{x^{2} - x - x^{2} + 6 x + 4 x + 4 \cdot - 6}{2} = 33$

Step 1.2.8.1.3

Multiply $4$ by $- 6$ .

$\frac{x^{2} - x - x^{2} + 6 x + 4 x - 24}{2} = 33$

Step 1.2.8.2

Add $6 x$ and $4 x$ .

$\frac{x^{2} - x - x^{2} + 10 x - 24}{2} = 33$

Step 1.3

Simplify terms.

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Step 1.3.1

Combine the opposite terms in $x^{2} - x - x^{2} + 10 x - 24$ .

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Step 1.3.1.1

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

$\frac{- x + 0 + 10 x - 24}{2} = 33$

Step 1.3.1.2

Add $- x$ and $0$ .

$\frac{- x + 10 x - 24}{2} = 33$

Step 1.3.2

Add $- x$ and $10 x$ .

$\frac{9 x - 24}{2} = 33$

Step 1.3.3

Factor $3$ out of $9 x - 24$ .

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Step 1.3.3.1

Factor $3$ out of $9 x$ .

$\frac{3 (3 x) - 24}{2} = 33$

Step 1.3.3.2

Factor $3$ out of $- 24$ .

$\frac{3 (3 x) + 3 (- 8)}{2} = 33$

Step 1.3.3.3

Factor $3$ out of $3 (3 x) + 3 (- 8)$ .

$\frac{3 (3 x - 8)}{2} = 33$

Step 2

Multiply both sides by $2$ .

$\frac{3 (3 x - 8)}{2} \cdot 2 = 33 \cdot 2$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{3 (3 x - 8)}{2} \cdot 2$ .

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Step 3.1.1.1

Cancel the common factor of $2$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{3 (3 x - 8)}{2} \cdot 2 = 33 \cdot 2$

Step 3.1.1.1.2

Rewrite the expression.

$3 (3 x - 8) = 33 \cdot 2$

Step 3.1.1.2

Apply the distributive property.

$3 (3 x) + 3 \cdot - 8 = 33 \cdot 2$

Step 3.1.1.3

Multiply.

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Step 3.1.1.3.1

Multiply $3$ by $3$ .

$9 x + 3 \cdot - 8 = 33 \cdot 2$

Step 3.1.1.3.2

Multiply $3$ by $- 8$ .

$9 x - 24 = 33 \cdot 2$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $33$ by $2$ .

$9 x - 24 = 66$

Step 4

Solve for $x$ .

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Step 4.1

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1.1

Add $24$ to both sides of the equation.

$9 x = 66 + 24$

Step 4.1.2

Add $66$ and $24$ .

$9 x = 90$

Step 4.2

Divide each term in $9 x = 90$ by $9$ and simplify.

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Step 4.2.1

Divide each term in $9 x = 90$ by $9$ .

$\frac{9 x}{9} = \frac{90}{9}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $9$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{90}{9}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{90}{9}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $90$ by $9$ .

$x = 10$