Prove the quadratic formula.

The quadratic formula is a method for finding the roots of a quadratic equation.

To derive the quadratic formula, we start with a general quadratic equation in the form ax^2 + bx + c = 0, where a, b, and c are constants. We can solve for x by completing the square:

ax^2 + bx + c = 0
ax^2 + bx = -c
a(x^2 + (b/a)x) = -c
a(x^2 + (b/a)x + (b/2a)^2) = -c + (b/2a)^2
a(x + b/2a)^2 = (b^2 - 4ac)/4a^2
x + b/2a = ±√((b^2 - 4ac)/4a^2)
x = (-b ± √(b^2 - 4ac))/2a

This is the quadratic formula. It gives us the two roots of the quadratic equation, which may be real or complex depending on the discriminant (b^2 - 4ac). If the discriminant is negative, the roots are complex conjugates. If the discriminant is zero, the roots are equal. If the discriminant is positive, the roots are real and distinct.

The quadratic formula is a powerful tool for solving quadratic equations, and it is used extensively in mathematics, science, and engineering. It allows us to find the roots of any quadratic equation quickly and easily, without having to complete the square or use other methods. It is also a fundamental result in algebra, and it is important for students to understand how it is derived and how it can be used.