Quadratic equations Lesson

See Also: Graphing Quadratic Calculator
See Also: Related Word Problems
See Also: Practice solving quadratics
See Also: In Depth Wikipedia article on Quadratic equation

Contents

Definitions
Standard Form
Quadratic Formula
Discussing Discriminant
Graphing Quadratics
Summary: Steps to solve quadratics

The quadratic Equation
ax²+bx+c=0
has solutions:

Typical plot of a quadratic
See a lesson on Graphing

Definitions

An exercise where you are asked to solve a quadratic equation is mostly a test of your attention span. If you can keep yourself focused for 5 minutes and perform all the steps that are clearly given to you, you win. You don't have to be smart or creative.

A quadratic equation is an equation like ax² + bx + c = 0, where a, b, and c are some numbers. These numbers are called coefficients. What you are typically required to do is find the values of x at which the left side of the equation (expression ax² + bx + c) is equal to zero.

Example: for equation x²-1 = 0, it so happens that the values of x that make x²-1 equal to zero are +1 and -1. This means that when you replace x with 1, the expression would be 1²-1 = 0. And it's obvious that this equation is true. When you use value -1 for x, you would get (-1)²-1 = 0, or after noting that -1 squared is 1, you again have 1-1 = 0, a true equation.

A frequent mistake that students make is not getting the "coefficients" right. They often forget a sign or confuse b with c. They they apply the correct formula to wrong numbers and get wrong results. Don't let this happen to you.

Equation	a	b	c
3x²-2x-1=0	3	-2	-1
x²-1=0	1	0	-1
-x²=0	-1	0	0
-x²-2x-3=0	-1	-2	-3
x²=1	This equation needs to be converted to standard form first.

Standard Form

Any equation that equates a second order polynomial to zero is a quadratic equation. But it has to be in a standard form in order for the standard formula to apply.The standard form is always, no exceptions, as follows:

ax²+bx+c = 0

Note that a, b, and c can be positive as well as negative. The right part of the equation should always be zero.

Quadratic Formula Proven

+x+=+%28-+b+%2B-+sqrt%28+b%5E2+-4ac+%29%29%2F2a%29

Click here to see the quadratic formula proven: (very educational)

In Depth Wikipedia article on completing the square

The +- sign used above is actually a shortcut: it means that there are two solutions, one with the sign + and another is with the sign -.

Discussing discriminant

If you look at the formula above, you will notice the really imprtant part of it inside the square root sign: it is called a discriminant. It is usually denoted as d:

d = b² - 4ac

If d is positive, that means that the expression under the square root is positive and therefore there are two distinct solutions.

If d is zero, this means that the expression under the square root sign is zero and therefore the value of the square root is zero too. This means that regardless of whether you add that zero or subtract it, you get the same result. So there is only one root.

If d is less than zero: as you know, there is no real number whose square is negative. Therefore, in such cases there are no real solutions.

This is discussed at length in the proof of the quadratic formula page.

Graphing Quadratics

Graphing quadratics is discussed at length in the lesson on Graphing. See also Parabola.

Summary: Steps to Solve Quadratics

You should always perform these steps when solving a quadratic equation. Note how these steps are followed in our quadratic equation calculator. If you miss even one of them, a mistake is almost guaranteed.

Remember that this is an exercise for your attention span.
Make sure that the equation is written in form ax²+bx+c = 0. Make sure that the right part is zero, for example.
Write down a, b, c. Make sure that you get the signs right.
Calculate the discriminant, d = b²-4ac. Again, do not forget about signs
If the discriminant is negative, stop. There is no real solution.
If the discriminant is not negative, calculate the solution according to the formula:
Practice solving quadratics here