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(x-1)^2=3x-5

2 min read Jun 17, 2024
(x-1)^2=3x-5

Solving the Quadratic Equation: (x-1)^2 = 3x - 5

This article will guide you through solving the quadratic equation (x-1)^2 = 3x - 5. We will explore the steps involved in finding the solutions, including expanding the equation, rearranging terms, and applying the quadratic formula.

Expanding and Rearranging

  1. Expand the left side of the equation: (x - 1)^2 = (x - 1)(x - 1) = x^2 - 2x + 1

  2. Rewrite the equation with all terms on one side: x^2 - 2x + 1 = 3x - 5 x^2 - 5x + 6 = 0

Solving the Quadratic Equation

We now have a standard quadratic equation in the form ax^2 + bx + c = 0, where a = 1, b = -5, and c = 6. To solve for x, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

  1. Substitute the values of a, b, and c into the formula: x = (5 ± √((-5)^2 - 4 * 1 * 6)) / (2 * 1)

  2. Simplify the expression: x = (5 ± √(25 - 24)) / 2 x = (5 ± √1) / 2 x = (5 ± 1) / 2

  3. Calculate the two possible solutions: x1 = (5 + 1) / 2 = 3 x2 = (5 - 1) / 2 = 2

Conclusion

Therefore, the solutions to the quadratic equation (x-1)^2 = 3x - 5 are x = 3 and x = 2. You can verify these solutions by substituting them back into the original equation.

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