What is a quadratic equation?

The a,b, and c here represent real number coefficients. So this means we are talking about an equation that is a constant times the variable squared plus a constant times the variable plus a constant equals zero, where the coefficient a on the variable squared can't be zero, because if it were then it would be a linear equation.

Examples

All these equations are equivalent to equations of the above form. The first one is already in that form. The second one can be put into it by subtracting 2x+3 from both sides. The third one can be put into it by multiplying out and then subtracting 5 from both sides.

Standard Form

The form

is the standard form for a quadratic equation, and for future reference, here the letter a will always mean the coefficient on the square of the variable, and b will be the coefficient on the variable, and c will be the constant term. To get a quadratic into standard form you must remove all parentheses and combine all like terms and add or subtract something from both sides so that the right side will be zero. Once you have your equation in standard form you can identify a,b, and c.

Example

Problem: Write the equation in standard form and identify a,b, and c.

Solving

Quadratic equations are harder to solve than linear equations, because once you have them in standard form it is hard to simplify them any further, and in this form there are still two occurrences of the variable, so it's hard to see what we can do to get the variable alone.

So we have to find some clever tricks to get around this problem.

Solving by Factoring

If A and B are real numbers and AB=0, then either A=0 or B=0.

This is a very special property that only zero has. For other numbers there are lots of ways to multiply and get them, but not for zero. For zero, the only way to multiply numbers and get it, is if one of the numbers is zero.

The principle of zero products allows us to reduce a complicated equation to simpler equations provided the right side of the equation is zero, and the equation is factored, because we can set each of the factors equal to zero.

• To solve a quadratic by factoring, first you must make sure it is in standard form. It is especially important that it is set equal to zero, because remember, the principle of zero products only works for zero.

• Then you must factor the left side.

• Then you set each of your factors equal to zero and solve the equations you get to find the solutions to your equation.

Example

What if you can't factor?

Consider the following equation.

This looks like a nice simple friendly equation, but we can't solve it by factoring, because we can't find two numbers to multiply and get -1 and add and get -2, so we are going to have to find another method.

But if only that minus sign on the 1 weren't there, then we could factor it really easily, in fact it would be a perfect square. How can we make that minus sign go away?

Well, one thing we could do is add 2 to both sides of the equation, and then the equation would become

Now, I know what you're saying. You are saying, "But you said that you have to get it set equal to 0 to solve it by factoring, because the principle of zero products only works for 0. What good does it do to have something factored and set equal to 2?"

And if you are saying this to yourself, you are absolutely right. But this isn't just any old factorization. It is aperfect square, and maybe you can do something with a perfect square set equal to 2.

If we could figure out how to solve equations like

that is, perfect squares set equal to numbers, then we could solve an equation like

And if we could find a way to add a number to both sides of other quadratics so that we can put them into the form perfect square equals constant, then maybe we could be able to solve them too.

This means that to help us solve quadratic equations, we need to learn two skills.

To work our way up to the task of solving equations of the form

let's first start with the slightly easier task of solving equation of the form

How do we solve an equation of the form

If x is greater than 0 then the obvious answer is

but this is not quite right because it only gives you the positive square root of d, and all positive numbers have two square roots, a positive one and a negative one. So to be sure that you are getting all solutions to an equation of this form, your answer must be

Now let's look at the more general equation of the form

This is really not much harder since anything you can do with x you should be able to do with x+k. x+k represents a number too. So solve for x+k and then add something to both sides of the equation to get x alone.

Example 1

Example 2

Completing the Square

This is what is meant by completing the square, and the secret to it is to expand out the expression

and see what makes perfect squares tick. Applying our formula for squaring a binomial, we get

The key here is to look at the relationship between the coefficient on x and the constant coefficient. The coefficient on x is 2k and the constant term is k². This means that if we know the coefficient on x, and we want to know what the constant term has to be for the expression to be a perfect square, then we need to divide the coefficient on x by 2 to get k, and then square to get k².

So if you have an expression of the form

and you want to find something to add to it to make it a perfect square, then you need to

Example

I hope the above has helped you understand the process of completing the square. If not, there is another approach to it that I have written an article about that you might find interesting for further understanding. It is a geometrical approach based on the method that many earlier mathematician used.

Solving by Completing the Square

One thing we left out. So far all of the equations we have solved have had a coefficient of 1 on x². What do we do if we have a coefficient other than 1 on there?

Well, we don't really have any method of completing the squares to deal with that situation, so the easiest thing to do is just divide both sides by it and put up with the fractions. With completing the squares, fractions are not so bad to deal with because there is no guess work.

Example

The Quadratic Formula

You see, with such a mechanical method like the method of completing the squares, why not just apply it to the general quadratic equation and solve all quadratics in the world at once, and be done with it, and never have to use algebra to solve a quadratic again.

Problem: Solve the equation:

Just do it the same way with the letters as you did with the numbers. First divide both sides by a. Then subtract c/a from both sides to be able to see more easily what to add to the left side to make it a perfect square. Then complete the square on the scratch paper and add what you get to both sides. Write the left side as the perfect square that it is. Instead of doing arithmetic on the right side you have to do a little bit of algebra, using 4a² as a common denominator. Then take square roots and subtract b/2a from both sides and use 2a a common denominator to get the final answer.

We have just solved all quadratics in the world at once and derived the quadratic formula, which says:

For any real numbers a,b, and c, the solutions to the equation

are

Solving by the Quadratic Formula

Since this formula is somewhat long and complicated, it is best to evaluate it in two smaller pieces by first evaluating the thing inside the radical,

and then put the result into the formula

The quantity

even has a name. It is called the discriminant. And there is another advantage to computing it first. Since it is what is in the radical, it can't be negative if there are going to be solutions to the equation, because you can't take a square root of a negative number, (unless you use the imaginary numbers, and we're not yet ready for them here) so if the discriminant comes out negative, then you don't have to do any more work, and all you have to do is write "no solution" on your paper and you are done. Sometimes you can determine this quite quickly by estimating, particularly if a and c are very large and b is small.

Example 1

Example 2

Imaginary Solutions

Once you know how to find square roots of negative numbers, you find imaginary solutions to quadratics by the completing the square or the quadratic formula pretty much like you find real ones. For the following examples the instruction is to solve the equation.

Example 1:

Solution:

Completing the Square:

Explanation:

Quadratic Formula:

Explanation:

Example 2:

Completing the Square:

Explanation:

Quadratic Formula:

Explanation:

Example 3:

Completing the Square:

Explanation:

Quadratic Formula:

Explanation: