The quadratic formula is a powerful tool used to solve quadratic equations that cannot be easily factored. It is derived from completing the square, and allows us to find the roots of any quadratic equation in the form ax^2 + bx + c = 0.

The quadratic formula is expressed as:

x = (-b ± √(b^2 - 4ac)) / (2a)

where a, b, and c are the coefficients of the quadratic equation.

Let's take an example to understand how to use the quadratic formula. Consider the equation 2x^2 - 7x + 3 = 0. Here, a = 2, b = -7, and c = 3. Plugging these values into the quadratic formula, we get:

x = (7 ± √( (-7)^2 - 4(2)(3) )) / (2(2))

Simplifying further, we have:

x = (7 ± √(49 - 24)) / 4

x = (7 ± √(25)) / 4

This gives us two solutions: x = (7 + 5) / 4 = 3 and x = (7 - 5) / 4 = 1/2.

The quadratic formula is extremely useful in solving quadratic equations that cannot easily be factored. It allows us to find the exact solutions, or roots, of the equation.

So next time you encounter a quadratic equation that cannot be factored easily, remember the quadratic formula and solve it like a pro!