(x%2B1)%2B3(4x-1)(4x%2B1)%3D2(5x-2)2-53.jpeg "(2x-7)(x+1)+3(4x-1)(4x+1)=2(5x-2)2-53")
Solving the Equation: (2x-7)(x+1)+3(4x-1)(4x+1)=2(5x-2)2-53
This article will guide you through the process of solving the given equation:
(2x-7)(x+1)+3(4x-1)(4x+1)=2(5x-2)2-53
Step 1: Expand the equation
First, we need to expand the equation by using the distributive property and simplifying the expressions:
- (2x-7)(x+1): This is a product of two binomials. We can expand it as follows: (2x-7)(x+1) = 2x² - 5x - 7
- 3(4x-1)(4x+1): This involves the product of two binomials, and a constant. Applying the difference of squares pattern (a²-b² = (a+b)(a-b)), we get: 3(4x-1)(4x+1) = 3(16x² - 1) = 48x² - 3
- 2(5x-2)²: This involves squaring a binomial. We can expand it as: 2(5x-2)² = 2(25x² - 20x + 4) = 50x² - 40x + 8
Now, the expanded equation looks like this:
2x² - 5x - 7 + 48x² - 3 = 50x² - 40x + 8 - 53
Step 2: Combine like terms
Next, we combine the terms with similar powers of x:
50x² - 5x - 10 = 50x² - 40x - 45
Step 3: Isolate the x variable
To isolate the x variable, we can subtract 50x² from both sides of the equation:
-5x - 10 = -40x - 45
Then, add 40x to both sides:
35x - 10 = -45
Finally, add 10 to both sides:
35x = -35
Step 4: Solve for x
Divide both sides by 35 to solve for x:
x = -1
Conclusion
Therefore, the solution to the equation (2x-7)(x+1)+3(4x-1)(4x+1)=2(5x-2)2-53 is x = -1.