A quadratic equation is a polynomial equation of the second degree, with the generalized form of ax²+bx+c=0.

By the fundamental theorem of algebra we know that this equation has two solutions, but we don’t know whether or not they are both real. Let’s explore one of the methods used for solving quadratic equations - the Quadratic Formula.

The quadratic formula is a method used to determine the roots of a quadratic equation, using a pre-derived solution to the equation. The formula is in terms of a, b and c, and looks like:

If you are wondering where this formula came from, take a quick look at the Quadratic Formula Derivation tutorial.

Ok, let’s put the formula to work. We are trying to solve an equation of the form ax²+bx+c=0 for x. Let’s take a look at an example:

x²+6=-5x

This equation appears to be quadratic, but to be sure we need to move all of the terms to the left-hand side of the equation, so that the right-hand side of the equation equals 0.

x²+5x+6=0

This equation is quadratic since it is in terms of a single variable and the equation is of the second degree.

Comparing the standard format of a quadratic equation to our example, we see:

ax²+bx+c=0

x²+5x+6=0

From this comparison, we can see that the coefficients of each term are the values a, b, and c in the formula. In this example a=1, b=5, and c=6. Now we can substitute these values into the quadratic formula to solve for x:

x = (-(5) +/- sqrt(5² -4(1)(6))) / ( 2(1))

Now let’s simplify to find the value(s) of x and the solution to the quadratic equation:

x = (-5 +/ - sqrt (25-24) )/ 2

x = (-5 +/- sqrt (1))/2

We must solve for both the + and –, so the final step is:

x = (-5 - 1)/2 = -6/2 = -3

x = (-5+1)/2 = -4/2 = -2

And that’s it. The solution to the quadratic equation x² +6=-5x is x=-3,-2.