Quadratic Equations

What is a quadratic equation?

A quadratic equation is an equation that can be written in this form.

ax²+bx+c=0

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

2x²+3x+1=0, x²+x=2x+3, (x+2)(x+3)=5

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

ax²+bx+c=0

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

This and many of the other examples below are from my MathHelp collection of problem sets, Quadratic Equations. For more practice and worked out examples for this or any other techniques explained here, click on the MathHelp link at the bottom of the page.

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

Solution: Multiply the left side out and then subtract the 5 from both sides.

Solving

Now lets talk about solving these equations.

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

One trick is to solve the equation by factoring. This trick works because of the principle of zero products. The principle of zero products says

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

Problem: Solve the equation. Solution: The second line here comes from setting x+1 and x+6 equal to 0 by the principle of zero factors.

What if you can't factor?

But some quadratics are difficult to factor, so for these equations we need other methods. The method of completing the square is a method that will work for any quadratic, but it is a little bit complicated, so I will introduce it slowly and step by step. But first to give you an overview of where we are going, I will show you a simple example of it.

Consider the following equation.

x²-2x-1=0

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

x²-2x+1=2 and this factors to (x-1)²=2

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 a perfect 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

(x-1)²=2

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

x²-2x-1=0.

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.
 

Solve equations of the form (x+k)2=d, where k and d are numbers.  Find a way of figuring out what number to add to both sides of a quadratic equations so that the left side will become a perfect square.

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

(x+k)²=d

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

x2=d

How do we solve an equation of the form

x2=d?

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

If x less than 0 then what happens? What kind of number can you square and get a negative number? If you square a positive number then clearly you get a positive number. But if you square a negative number then you have a product of two negative numbers, so you still get a positive number. So what is left for squaring and getting a negative number? Nothing. So the equation has no solutions.

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

(x+k)²=d

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

Problem: Solve the equation. Solution:

Example 2

Problem: Solve the equation. Solution: Anything you can do with x, you can do with x+1, and once you find x+1 all you have to do to get x is subtract 1.

Completing the Square

Now to problem number two, that of finding something to add to a quadratic to make it a perfect square.

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

(x+k)²

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

(x+k)²=x²+2kx+k²

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

x²+bx

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

Divide b by 2 to get k  Square k to get k2.

Example

Problem: Complete the square. Solution: The yellow part is the scratch paper. On the scratch paper you first divide the 9 by 2 and then square the result. Don't worry about the minus sign, because it will go away when you square anyway. Then the number you get will be the number you need to add to the expression to make a perfect square out of it. After you do that it is good practice to write it as the square that it is. For that you can use the first line of your scratch paper and match the sign with the sign of the second term of the original expression.

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. You can read my article A Geometrical Approach to Completing the Square to find out about it.

Solving by Completing the Square

Now we are ready to use the method of completing the squares to solve quadratic equations. The best way to do this is as follows.
 

Add something to both sides so that the left side has no constant term. Figure out what to add to the left side to make it a perfect square, and add that to both sides. Write the left side as the perfect square that it is and do the arithmetic on the right side. Solve the equation you get by the methods of equations of the form (x+k)2=d

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

Problem: Solve the equation. Solution: First we divide both sides of the equation by 2, to get a coefficient of 1 on the first term. 0/2 is still 0. Then we add 1/2 to both sides so that it is easier to figure out what to add to make the left side a perfect square. Then we complete the square in order to figure out what to add to both sides to make the left side a perfect square. The scratch work for this is shown in yellow. The best way to divide a fraction by 2 is to multiply it by 1/2. Then we write the left side as the perfect square that it is, and do the arithmetic on the right side. Now take square roots and add 3/4 to both sides to get the final answer.

The Quadratic Formula

Now that you have learned the method of completing the squares, I will tell you a secret. The methods of completing the squares is such a good method for solving quadratics that it is very seldom used for it. It is used for other things in mathematics. See my article  Equations of Circles  for another use for it. But for solving quadratic equations, it is such a good method that it puts itself out of business.

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:

Solution:

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

ax²+bx+c=0

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,

b²-4ac

and then put the result into the formula

The quantity

b²-4ac

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

Problem: Solve the equation. Solution:

Example 2

Problem: Solve the equation. Solution: No solution. b is -2, which is very small in comparison to a=6 and c=27, so you don't even have to compute the discriminant to see that it is going to be negative.

Imaginary Solutions

This section is for more advanced students who know about imaginary numbers. For a brief introduction see my article Complex Number Arithmetic. If you know about imaginary numbers, you don't have to stop when you see the square root of a negative number, because with imaginary numbers you can take the square root of a negative number. To find the square root of any negative number you just take the square root of the corresponding positive number and multiply it by i, the square root of -1. This makes sense at least once you believe in the idea that the square root of -1 is i, because of the multiplication rule for square roots. (See Square Roots.) (It is customary usually to write the real number after the i when it is a square root so that it is clear that the i is not inside the radical.)

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:

x²+2x+5=0

Solution:

Completing the Square:

Explanation:

First we add -5 to both sides to get the constant on the right side of the equation so that it is more clear what we need to add to the left side to make it a perfect square. Then we complete the square. 2/2=1, 1²=1, so 1 is the number we add to the left side to get a perfect square. Whatever you do to one side of an equation, you have to do to the other side, so we also add 1 to the right side of the equation. Then we write the left side as the perfect square that it is, and do the arithmetic on the right side. In this equation we get a negative number on the right side, but with imaginary numbers we can deal with that. The two square roots of -4 are 2i and -2i, so x+1 has to be one of them. Then to find out what x is, all we have to do is add -1 to both sides.

Quadratic Formula:

Explanation:

First compute the discriminant and find its square root. The square root of -16 is the square root of 16 times i, 4i. Then we just fit in the square root in its place in the formula. The numerator has a common factor of 2 that we can factor out and cancel with the 2 in the denominator.

Example 2:

x²+x+1=0

Completing the Square:

Explanation:

Again we first subtract 1 from both sides so that it is more clear what we need to add. For the completing the square part, this time the coefficient on x is 1. Half of 1 is 1/2 and (1/2)²=1/4, so 1/4 is the completing the square number. We add it to both sides, because anything you add to one side you have to add to the other side. Then again we get a negative number to take plus and minus square roots of, but we can handle that with imaginary numbers. We just break it up into square roots of -1, 3, and 4. The square root of -1 is i, the square root of 4 is 2, and the square root of 3 is the square root of 3. Then to get the final answer we add -1/2 to both sides.

Quadratic Formula:

Explanation:

First we figure out the discriminant, which comes out to be -3, and then we take its square root. The square root of a negative number is just the square root of the corresponding positive one times i. This time we can't the square root of 3 is an irrational number, so it is better left undone thinking of radical 3 is the name for the exact number that you multiply by itself to get 3. Then after we have found the square root of -3, we can put it in place of the radical in the formula to get our final answer.

Example 3:

3x²+2x+3=0

Completing the Square:

Explanation:

In this one there is a coefficient other than 1 on the x², namely 3, so first we have to get rid of it by dividing both sides by 3. Then from there it is pretty much like the other examples. Next we add -1 to both sides to see more clearly what we need to add to make the left side a perfect square. The coefficient on x is 2/3. To find half of 2/3 we multiply it by 1/2 and get 1/3. Then to find the completing the square number, we square that and get 1/9, which we add to both sides of the equation. Then we write the left side as the perfect square that it is and do the arithmetic on the right side. And again since this section is about imaginary solutions, we get a negative number on the right side of the equation to take the square root of. Then after taking plus and minus square roots, we add -1/3 to both sides to get the final answer.

Quadratic Formula:

Explanation:

First we find the discriminant, and we get -32. The square root of -32 is the square root of -1 times the square root of 16 times the square root of 2. The square root of -1 is i, the square root of 16 is 4, and the square root of 2 is something nasty and irrational, so it is left as the square root of 2. Then we put this square root in place of the radical in the formula and simplify to get the final answer.