I will illustrate and describe how to find the roots of a quadratic equation using the quadratic formula. I will also include some extra information, such as the discriminant of the quadratic formula and the proof of the quadratic formula. Enjoy!
Let’s start with the definition of a quadratic equation.
A quadratic equation is an equation in the form ax²+bx+c=0 in which a does not equal 0 and x is an unknown variable. Let’s say our quadratic equation is (1)x²+4x+4=0, and we want to find the roots of the equation. First of all, we must know what a root is. A root is the point where the graph of the equation meets the x-axis. We could graph it out. You could substitute different values for x and find the spot(s) where it meets the x-axis. Or, we could use the quadratic function. The quadratic function is generally denoted as the following:
Visually, it’s like:
So, If we were to find the roots of the equation x²+4x+4, we would do the following.
If you were to replace the x in the equation with -2 (edit by author), you would get a true equation. This is a very simple case. Usually, you would get a number under the square root sign, indicating one out of two things. If the number under the square root sign is negative, then your equation does not have any roots. If the number under the square root sign is positive, your equation has two roots. Each root can be found by replacing “+” or “-” for the “±” sign. If the number under the square root sign is 0, as in this case, then your equation has one solution.
Discriminant of the Quadratic Formula
The discriminant of a quadratic formula is the polynomial/expression under the square root sign, which is,
You can use this to find the number of solutions to a quadratic equation. If the discriminant is greater than 0, then the quadratic equation has 2 solutions. For example, x²+5x+6, 25-4(1)(6)=1, which is greater than 0. If the discriminant is 0, then the quadratic equation has 1 solution. For example, the example first given, or x²+6x+9, 36-4(1)(9)=0, which is equal to 0. If the discriminant is less than 0, then the quadratic equation has 0 real roots (solutions), as in 2x²+4x+5, 16-4(2)(5)=-24, which is less than 0.
Let’s do another example:2x²+5x+2