In this lesson, our instructor Vincent Selhorst-Jones will teach you about Completing the Square and the Quadratic Formula. Youll learn how to use square roots to find the value of x, how to make complex polynomials easy to solve, and how to complete the square. Vincent will also teach you how to follow format to use formulas, how to use the quadratic formula, and how the discriminant works. The lesson ends with four practice examples.
In this lesson, we will be working just with quadratic polynomials: polynomials with degree 2. Thus, quadratics are of the form
ax2 + bx + c,
where a, b, c are constant real numbers and a ≠ 0.
Whenever we take the square roots on both sides of an equation when doing algebra, we put a ± on one side. Example: x2 = 4 ⇒ x = ±2.
To complete the square for any quadratic, we want to put it in the form
()2 − k.
Once in this form, we can easily set it to 0 and solve.
We convert as follows:
ax2 + bx + c = 0 ⇔
b2 − 4ac
Don't try to memorize the formula above, instead, watch the video and learn the general method behind it. While the formula will work, it's very difficult to remember. It's much easier to learn the step-by-step method to produce it.
From this conversion, we can create the quadratic formula: a formula that gives an easy way to solve for the roots of any quadratic polynomial.
To use the quadratic formula above, the polynomial must be set up in the format ax2 + bx + c = 0. The quadratic must be put into that format before you can use the formula.
There are three possible numbers of roots for a quadratic to have: 2, 1, or 0. We determine this number from the discriminant (contained in the quadratic formula): b2−4ac. This value tells us how many roots the polynomial has:
b2 − 4ac > 0 ⇒ 2 roots;
b2 − 4ac = 0 ⇒ 1 root;
b2 − 4ac < 0 ⇒ 0 roots.
Completing the Square and the Quadratic Formula
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.