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Solving the Quadratic Equation: (x+2)^2 = 9(x^2-4x+4)
This article will guide you through the steps of solving the quadratic equation (x+2)^2 = 9(x^2-4x+4).
Expanding and Simplifying
First, we need to expand both sides of the equation:
- Left side: (x+2)^2 = (x+2)(x+2) = x^2 + 4x + 4
- Right side: 9(x^2 - 4x + 4) = 9x^2 - 36x + 36
Now our equation becomes: x^2 + 4x + 4 = 9x^2 - 36x + 36
Rearranging the Equation
To solve for x, we need to rearrange the equation into the standard quadratic form (ax^2 + bx + c = 0). Subtract x^2, 4x, and 4 from both sides:
0 = 8x^2 - 40x + 32
Solving the Quadratic Equation
We now have a quadratic equation in standard form. We can solve this using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Where a = 8, b = -40, and c = 32.
Substituting the values into the quadratic formula:
x = (40 ± √((-40)^2 - 4 * 8 * 32)) / (2 * 8) x = (40 ± √(1600 - 1024)) / 16 x = (40 ± √576) / 16 x = (40 ± 24) / 16
This gives us two solutions:
- x1 = (40 + 24) / 16 = 64 / 16 = 4
- x2 = (40 - 24) / 16 = 16 / 16 = 1
Conclusion
Therefore, the solutions to the quadratic equation (x+2)^2 = 9(x^2-4x+4) are x = 4 and x = 1.