%5E2-4x%5E2%3D0.jpeg "(2x-3)^2-4x^2=0")
Solving the Quadratic Equation: (2x-3)^2 - 4x^2 = 0
This article will guide you through the steps of solving the quadratic equation (2x-3)^2 - 4x^2 = 0. We will explore the different methods of solving this equation, including:
1. Expanding and Simplifying
-
Step 1: Expand the square: (2x-3)^2 = (2x-3)(2x-3) = 4x^2 - 12x + 9
-
Step 2: Substitute the expanded term back into the original equation: 4x^2 - 12x + 9 - 4x^2 = 0
-
Step 3: Simplify the equation: -12x + 9 = 0
-
Step 4: Solve for x: -12x = -9 x = -9/-12 x = 3/4
Therefore, the solution to the equation (2x-3)^2 - 4x^2 = 0 is x = 3/4.
2. Using the Difference of Squares Formula
-
Step 1: Recognize the difference of squares pattern: The equation can be rewritten as [(2x-3) + 2x][(2x-3) - 2x] = 0
-
Step 2: Apply the difference of squares formula: (4x-3)(-3) = 0
-
Step 3: Solve for x: 4x-3 = 0 or -3 = 0 x = 3/4 or No solution
This method also leads us to the solution x = 3/4.
Conclusion
We have successfully solved the quadratic equation (2x-3)^2 - 4x^2 = 0 using two different methods. Both methods confirm that the only solution to this equation is x = 3/4.
Understanding the different methods for solving quadratic equations is crucial in mathematics. By applying these techniques, you can confidently tackle various algebraic problems and find solutions for equations of varying complexity.