(x-6)^2+2x(x-6)=0

3 min read Jun 17, 2024
(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.

Related Post