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Solving the Quadratic Equation: (x-2)^2 - 5(x-2) = 0
This article will guide you through solving the quadratic equation (x-2)^2 - 5(x-2) = 0. We will explore different methods, including factoring and the quadratic formula, to find the solutions.
Understanding the Equation
The equation is a quadratic equation because the highest power of the variable 'x' is 2. It can be rewritten in standard quadratic form:
(x-2)^2 - 5(x-2) = 0
=> x^2 - 4x + 4 - 5x + 10 = 0
=> x^2 - 9x + 14 = 0
Solving by Factoring
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Factor the quadratic expression:
We need to find two numbers that add up to -9 (the coefficient of the x term) and multiply to 14 (the constant term). These numbers are -7 and -2.So, we can factor the equation as: (x - 7)(x - 2) = 0
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Set each factor equal to zero:
- x - 7 = 0 => x = 7
- x - 2 = 0 => x = 2
Therefore, the solutions to the equation are x = 7 and x = 2.
Solving using the Quadratic Formula
The quadratic formula provides a general solution for any quadratic equation in the form ax^2 + bx + c = 0.
x = (-b ± √(b^2 - 4ac)) / 2a
In our equation: a = 1, b = -9, and c = 14
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Substitute the values into the formula: x = (9 ± √((-9)^2 - 4 * 1 * 14)) / (2 * 1)
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Simplify: x = (9 ± √(81 - 56)) / 2 x = (9 ± √25) / 2 x = (9 ± 5) / 2
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Solve for x:
- x = (9 + 5) / 2 = 7
- x = (9 - 5) / 2 = 2
Again, we find the solutions are x = 7 and x = 2.
Conclusion
We have successfully solved the quadratic equation (x-2)^2 - 5(x-2) = 0 using two different methods: factoring and the quadratic formula. Both methods lead to the same solutions, demonstrating the versatility of these techniques in solving quadratic equations.