%5E2%2B2x(x-6)%3D0.jpeg "(x-6)^2+2x(x-6)=0")
Solving the Quadratic Equation: (x-6)^2 + 2x(x-6) = 0
This article will guide you through solving the quadratic equation (x-6)^2 + 2x(x-6) = 0. We will utilize methods like factoring and the quadratic formula to find the solutions.
1. Simplifying the Equation
Firstly, let's simplify the equation by expanding the squares and multiplying the terms:
(x - 6)(x - 6) + 2x(x - 6) = 0 x^2 - 12x + 36 + 2x^2 - 12x = 0 3x^2 - 24x + 36 = 0
2. Factoring the Equation
Now, we can factor out a common factor of 3 from the equation:
3(x^2 - 8x + 12) = 0
Next, factor the quadratic expression inside the parentheses:
3(x - 2)(x - 6) = 0
3. Finding the Solutions
To find the solutions, we set each factor equal to zero:
- 3 = 0 (This is not a valid solution)
- x - 2 = 0 => x = 2
- x - 6 = 0 => x = 6
Therefore, the solutions to the quadratic equation (x-6)^2 + 2x(x-6) = 0 are x = 2 and x = 6.
4. Using the Quadratic Formula
Alternatively, we can use the quadratic formula to solve for x:
x = [-b ± √(b^2 - 4ac)] / 2a
Where a = 3, b = -24, and c = 36.
Plugging in these values, we get:
x = [24 ± √((-24)^2 - 4 * 3 * 36)] / (2 * 3) x = [24 ± √(576 - 432)] / 6 x = [24 ± √144] / 6 x = [24 ± 12] / 6
This gives us two possible solutions:
x = (24 + 12) / 6 = 6 x = (24 - 12) / 6 = 2
As expected, we obtain the same solutions as through factoring.
Conclusion
We have successfully solved the quadratic equation (x-6)^2 + 2x(x-6) = 0 using both factoring and the quadratic formula. Both methods lead to the same solutions: x = 2 and x = 6.