(x%2B10)%2F3-(x%2B4)(x%2B10)%2F12%3D(x-2)(x%2B4)%2F4.jpeg "(x-2)(x+10)/3-(x+4)(x+10)/12=(x-2)(x+4)/4")
Solving the Equation: (x-2)(x+10)/3 - (x+4)(x+10)/12 = (x-2)(x+4)/4
This article will guide you through the steps of solving the equation: (x-2)(x+10)/3 - (x+4)(x+10)/12 = (x-2)(x+4)/4
1. Finding a Common Denominator
To start, we need to find a common denominator for all fractions in the equation. The least common multiple of 3, 12, and 4 is 12.
- Multiply the first fraction by 4/4: (4/4) * (x-2)(x+10)/3 = 4(x-2)(x+10)/12
- The second fraction already has a denominator of 12.
- Multiply the third fraction by 3/3: (3/3) * (x-2)(x+4)/4 = 3(x-2)(x+4)/12
Now our equation looks like this: 4(x-2)(x+10)/12 - (x+4)(x+10)/12 = 3(x-2)(x+4)/12
2. Simplifying the Equation
Since all fractions have the same denominator, we can combine the numerators:
4(x-2)(x+10) - (x+4)(x+10) = 3(x-2)(x+4)
3. Expanding the Equation
Next, we need to expand the products on both sides of the equation:
4(x^2 + 8x - 20) - (x^2 + 14x + 40) = 3(x^2 + 2x - 8)
4. Simplifying Further
Now, distribute the constants and simplify:
4x^2 + 32x - 80 - x^2 - 14x - 40 = 3x^2 + 6x - 24
5. Combining Like Terms
Combine like terms on both sides of the equation:
3x^2 + 18x - 120 = 3x^2 + 6x - 24
6. Isolating the Variable
Subtract 3x^2 from both sides:
18x - 120 = 6x - 24
Subtract 6x from both sides:
12x - 120 = -24
Add 120 to both sides:
12x = 96
7. Solving for x
Divide both sides by 12:
x = 8
Therefore, the solution to the equation (x-2)(x+10)/3 - (x+4)(x+10)/12 = (x-2)(x+4)/4 is x = 8.