(x−2)2−6(x−2)+5=0

4 min read Jun 17, 2024

Solving the Quadratic Equation: (x−2)² - 6(x−2) + 5 = 0

This article will guide you through the process of solving the quadratic equation (x−2)² - 6(x−2) + 5 = 0. We will explore different methods to find the solutions for x.

Understanding the Equation

The equation (x−2)² - 6(x−2) + 5 = 0 is a quadratic equation in standard form. We can see this by recognizing that it can be written as:

ax² + bx + c = 0

Where:

a = 1 (the coefficient of the x² term)
b = -8 (the coefficient of the x term)
c = 9 (the constant term)

This form highlights the key elements of a quadratic equation.

Solving by Factoring

One way to solve this equation is by factoring:

Recognize the pattern: Notice that the equation has a repeated term (x - 2) in both the first and second terms. This pattern suggests a substitution for simplification.
Substitution: Let y = (x - 2). Substituting this into the equation, we get: y² - 6y + 5 = 0
Factoring the simplified equation: This equation can now be factored easily: (y - 1)(y - 5) = 0
Solving for y: Setting each factor equal to zero, we get:
- y - 1 = 0 => y = 1
- y - 5 = 0 => y = 5
Substituting back for x: Now, substitute back (x - 2) for y to find the solutions for x:
- x - 2 = 1 => x = 3
- x - 2 = 5 => x = 7

Therefore, the solutions to the quadratic equation (x−2)² - 6(x−2) + 5 = 0 are x = 3 and x = 7.

Solving by the Quadratic Formula

The quadratic formula is a general method for solving quadratic equations in the standard form ax² + bx + c = 0.

The quadratic formula is: x = (-b ± √(b² - 4ac)) / 2a

Identify a, b, and c: From our original equation, we know a = 1, b = -8, and c = 9.
Substitute the values: Plug these values into the quadratic formula: x = (8 ± √((-8)² - 4 * 1 * 9)) / (2 * 1)
Simplify: x = (8 ± √(64 - 36)) / 2 x = (8 ± √28) / 2 x = (8 ± 2√7) / 2
Final Solutions: x = 4 ± √7

Therefore, the solutions to the quadratic equation (x−2)² - 6(x−2) + 5 = 0 are x = 4 + √7 and x = 4 - √7.

Conclusion

We have demonstrated two methods to solve the quadratic equation (x−2)² - 6(x−2) + 5 = 0. Both factoring and the quadratic formula lead to the same solutions, emphasizing the versatility of these methods in solving quadratic equations. The choice of method depends on the specific equation and the solver's preference.