(x-2)%2F10%2B2%2F5%3D(x-2)%5E2%2F5%2B(x-1)(x-3)%2F2.jpeg "(7x+1)(x-2)/10+2/5=(x-2)^2/5+(x-1)(x-3)/2")
Solving the Equation: (7x+1)(x-2)/10+2/5=(x-2)^2/5+(x-1)(x-3)/2
This article will guide you through the steps of solving the given equation.
Step 1: Find a Common Denominator
The first step is to find a common denominator for all the fractions in the equation. The least common multiple of 10, 5, and 2 is 10.
- Multiply the first term by 1/1:
(7x+1)(x-2)/10 * 1/1 = (7x+1)(x-2)/10 - Multiply the second term by 2/2:
2/5 * 2/2 = 4/10 - Multiply the third term by 2/2:
(x-2)^2/5 * 2/2 = 2(x-2)^2/10 - Multiply the fourth term by 5/5:
(x-1)(x-3)/2 * 5/5 = 5(x-1)(x-3)/10
Now the equation becomes:
(7x+1)(x-2)/10 + 4/10 = 2(x-2)^2/10 + 5(x-1)(x-3)/10
Step 2: Simplify the Equation
Since all the terms now have the same denominator, we can eliminate the denominators and work with the numerators only.
(7x+1)(x-2) + 4 = 2(x-2)^2 + 5(x-1)(x-3)
Step 3: Expand and Simplify
Expand the products and simplify the equation.
7x^2 - 13x - 2 + 4 = 2x^2 - 8x + 8 + 5x^2 - 20x + 15
Combining like terms:
7x^2 - 13x + 2 = 7x^2 - 28x + 23
Step 4: Solve for x
Subtract 7x^2 from both sides and simplify:
-13x + 2 = -28x + 23
Add 28x to both sides and simplify:
15x + 2 = 23
Subtract 2 from both sides and simplify:
15x = 21
Divide both sides by 15 and simplify:
x = 21/15 = 7/5
Solution
Therefore, the solution to the equation (7x+1)(x-2)/10+2/5=(x-2)^2/5+(x-1)(x-3)/2 is x = 7/5.