Solve for x (x(x-1))/2-((x-4)(x-6))/2=33

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

Step 1

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

Combine the numerators over the common denominator.

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

Simplify each term.

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

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

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $- 4$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 6$ by $- 1$ .

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

Multiply $4$ by $- 6$ .

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

Add $6 x$ and $4 x$ .

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

Simplify terms.

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

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

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

Add $- x$ and $0$ .

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

Add $- x$ and $10 x$ .

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

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

Factor $3$ out of $9 x$ .

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

Factor $3$ out of $- 24$ .

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

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.

Simplify the left side.

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Apply the distributive property.

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

Multiply.

Multiply $3$ by $3$ .

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

Multiply $3$ by $- 8$ .

$9 x - 24 = 33 \cdot 2$

Simplify the right side.

Multiply $33$ by $2$ .

$9 x - 24 = 66$

Step 4

Solve for $x$ .

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

Add $24$ to both sides of the equation.

$9 x = 66 + 24$

Add $66$ and $24$ .

$9 x = 90$

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

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

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

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $90$ by $9$ .

$x = 10$

Solve for x (x(x-1))/2-((x-4)(x-6))/2=33

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