%5E2-(3x%2B2)(3x-2)%3D(7x-1)(x%2B2)%2B(2x%2B1)%5E2-(4x%5E2%2B7).jpeg "(4x-1)^2-(3x+2)(3x-2)=(7x-1)(x+2)+(2x+1)^2-(4x^2+7)")
Solving the Equation: (4x-1)^2-(3x+2)(3x-2)=(7x-1)(x+2)+(2x+1)^2-(4x^2+7)
This article will guide you through the steps of solving the given algebraic equation:
(4x-1)^2-(3x+2)(3x-2)=(7x-1)(x+2)+(2x+1)^2-(4x^2+7)
Step 1: Expand the Expressions
Begin by expanding the squares and the products using the distributive property or the appropriate formulas:
- (4x-1)^2: (4x-1)(4x-1) = 16x^2 - 8x + 1
- (3x+2)(3x-2): This is a difference of squares pattern: (3x)^2 - (2)^2 = 9x^2 - 4
- (7x-1)(x+2): 7x^2 + 14x - x - 2 = 7x^2 + 13x - 2
- (2x+1)^2: (2x+1)(2x+1) = 4x^2 + 4x + 1
Now the equation becomes:
16x^2 - 8x + 1 - (9x^2 - 4) = 7x^2 + 13x - 2 + 4x^2 + 4x + 1 - (4x^2 + 7)
Step 2: Simplify by Removing Parentheses
Be careful with the negative signs in front of the parentheses:
16x^2 - 8x + 1 - 9x^2 + 4 = 7x^2 + 13x - 2 + 4x^2 + 4x + 1 - 4x^2 - 7
Step 3: Combine Like Terms
Combine all the x^2 terms, x terms, and constant terms:
7x^2 - 8x + 5 = 7x^2 + 17x - 8
Step 4: Isolate the Variable
To get all the x terms on one side, subtract 7x^2 from both sides:
-8x + 5 = 17x - 8
Next, subtract 17x from both sides:
-25x + 5 = -8
Step 5: Solve for x
Finally, subtract 5 from both sides and then divide by -25:
-25x = -13
x = 13/25
Therefore, the solution to the equation is x = 13/25.