Sure, let's solve the given quadratic equation using various methods:
1. Solve the equation by taking the square root on both sides:
x^2 = (2x - 9)^2
Since both sides are squared, one could think to solve it by taking the square root of both sides. But as squares hide the negative roots, it's better to transit to another way of solving this equation, expanding brackets and making equation in common quadratic form.
2. Solve the equation by expanding the brackets:
x^2 = (2x - 9)^2
x^2 = 4x^2 - 36x + 81
To bring it to the standard form of a quadratic equation, we need to bring all the terms to one side of the equation:
0 = 3x^2 - 36x + 81
This is a more common form for quadratic equations: ax^2 + bx + c = 0. In this case, a = 3, b = -36, and c = 81.
3. Solve the equation by factoring:
Now that we have the equation in standard form, we can factor it:
0 = 3(x^2 -12x + 27)
Factorization didn't simplify it so this path doesn't help us in this case and we thus move forward to another method - quadratic formula.
4. Solve the equation by using the quadratic formula:
The solutions for a quadratic equation in the standard form ax^2 + bx + c = 0 can be found using the quadratic formula, which is:
x = [-b ± sqrt(b^2 - 4ac)] / 2a
We substitute a = 3, b = -36, and c = 81 into the quadratic formula:
x = [36 ± sqrt((-36)^2 - 4*3*81)] / 2*3
x = [36 ± sqrt(1296 - 972)] / 6
x = [36 ± sqrt(324)] / 6
x = [36 ± 18] / 6
Therefore, the solutions to the quadratic equation are x = 3 and x = 9.