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Solving the Quadratic Equation: (x-2)^2 - 9 = 0
This article will guide you through the steps to solve the quadratic equation (x-2)^2 - 9 = 0.
Understanding the Equation
The equation (x-2)^2 - 9 = 0 is a quadratic equation in standard form. It can be rewritten as:
x^2 - 4x - 5 = 0
This equation represents a parabola, and solving it means finding the x-values where the parabola intersects the x-axis.
Methods of Solving
There are several methods to solve quadratic equations, including:
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Factoring:
- Factor the quadratic expression: (x - 5)(x + 1) = 0
- Set each factor equal to zero:
- x - 5 = 0 => x = 5
- x + 1 = 0 => x = -1
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Quadratic Formula:
- The quadratic formula provides the solutions for any quadratic equation in the form ax^2 + bx + c = 0:
x = (-b ± √(b^2 - 4ac)) / 2a
- In our case, a = 1, b = -4, and c = -5. Substitute these values into the quadratic formula:
x = (4 ± √((-4)^2 - 4 * 1 * -5)) / (2 * 1)
x = (4 ± √(36)) / 2
x = (4 ± 6) / 2
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This gives us two solutions:
- x = (4 + 6) / 2 = 5
- x = (4 - 6) / 2 = -1
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Completing the Square:
- Move the constant term to the right side: (x - 2)^2 = 9
- Take the square root of both sides: x - 2 = ±3
- Solve for x:
- x = 2 + 3 = 5
- x = 2 - 3 = -1
Solutions
Regardless of the method used, the solutions to the equation (x-2)^2 - 9 = 0 are:
x = 5 and x = -1
Conclusion
We have successfully solved the quadratic equation (x-2)^2 - 9 = 0 using three different methods. All methods lead to the same solutions: x = 5 and x = -1. These solutions represent the x-intercepts of the parabola represented by the equation.