Find the Roots of a Quadratic Equation

A quadratic equation is any equation in the form ax2 + bx + c = 0 where a ≠ 0. Though "finding the roots of a quadratic equation" may sound intimidating, it's not — "finding the roots" is just the same thing as solving the equation for x! Any quadratic equation can be solved with the formula x = (-b +/-√(b2 - 4ac))/2a. In addition, depending on the equation you're dealing with, there are several other tricks that can be used to help find the roots.

Steps

Using the Quadratic Formula

  1. Write your equation in the quadratic form. The official definition of a quadratic equation is a second-order polynomial equation expressed in a single variable, x, with a ≠ 0.[1] In simple terms, this just means that it's an equation with one variable (usually x) where the highest exponent of the variable is 2. In general terms, we can write this as ax2 + bx + c = 0
    • To get an equation in quadratic form, just get all of the terms on one side of the equals sign so that you have 0 on the other side. For example, if we want to get the equation 2x2 + 8x = -5x2 - 11 in quadratic form, we can do it like this:
    • 2x2 + 8x = -5x2 + 11
    • 2x2 + 5x2 + 8x = + 11
    • 2x2 + 5x2 + 8x - 11 = 0
    • 7x2 + 8x - 11 = 0 . Notice that this is in the standard ax2+ bx + c = 0 form mentioned above.
  2. Plug a, b, and c into x = (-b +/-√(b2 - 4ac))/2a. Finding the roots of a quadratic equation with the quadratic formula is easy - just use plug a, b, and c into the formula and solve for x! Since the form of a quadratic equation is ax2+ bx + c = 0, this means the number next to the x2 term is a, the number next to the x term is b, and the number without an x term is c.
    • For our example equation, 7x2 + 8x - 11 = 0, a = 7, b = 8, and c = -11.
    • Plugging this into the formula, we get x = (-8 +/-√(82 - 4(7)(-11)))/2(7)
  3. Solve. Once you've plugged your a, b, and c values into your formula, solving is just a matter of doing basic algebra operations until you get to the +/- symbol. We'll deal with that in the next step.
    • In our example, we'd solve like this:
    • x = (-8 +/-√(82 - 4(7)(-11)))/2(7)
    • x = (-8 +/-√(64 - (28)(-11)))/(14)
    • x = (-8 +/-√(64 - (-308)))/(14)
    • x = (-8 +/-√(372))/(14)
    • x = (-8 +/- 19.29/(14) . Let's stop here for now.
  4. Add and subtract to get two final answers. One of the tricky things about finding the roots of a quadratic equation is that you'll usually get two correct answers (if you're doing quadratic equations for school work, don't forget to list both to get full points!) To get both answers, finish solving the equation for x, once using a + and once using a -.
    • Adding, we get:
    • x = (-8 + 19.29)/(14)
    • x = 11.29/14
    • x = 0.81
    • Subtracting, we get:
    • x = (-8 - 19.29)/(14)
    • x = (-27.29)/(14)
    • x = -1.95 .
    • Thus, our answers are 0.81 and -1.95.
  5. Check your answers. If you have time, it's a good idea to check the roots of your quadratic equation once you've found them. Since solving a quadratic equation involves doing a long string of math operations, it's easy to make simple mistakes that can affect your answers. Luckily, the easy checking methods below should reveal whether or not you've got the right roots.
    • The quickest, easiest way to check your answer is to simply plug your a, b, and c terms into an automatic quadratic solving program. These are easily found online — for instance, here is one from mathisfun.com.[2]
  6. Alternatively, check your answers manually. If you're in a situation where you can't use a handy online tool to check your answers, you can still see whether you have the correct roots by plugging them in for x in your original equation. If your equation comes out to zero (or very close to it — this is usually due to rounding), you have the correct roots.
    • Let's plug our answers back in to 7x2 + 8x - 11 = 0 to see if they are correct:
    • 7(-1.95)2 + 8(-1.95) - 11
    • 26.62 - 15.6 - 11
    • 26.62 - 26.5 = 0.02 — this is almost zero, so the difference is probably from rounding and not from having the wrong answer.
    • 7(0.81)2 + 8(0.81) - 11
    • 4.59 + 6.48 - 11 = 0.07 — see above. Our answers are most likely correct.

Finding Roots by Factoring

Factoring With an "A" Value of 1

  1. Start with an equation in the quadratic form. Though the quadratic formula described above is a valuable tool, it isn't the only way to solve quadratic equations. For example, some quadratic equations can be factored, which involves rewriting them in a way that makes them easy to solve. To start, however, you'll want to have your equation in standard quadratic form: ax2 + bx + c = 0.
    • In this section, we'll only be dealing with quadratics that have an "a" variable equal to 1. If the a variable is not 1, the process is a little harder (see below.) Let's use x2 + 7x + 12 = 0 as our example equation in this section. In the next few steps, we'll factor and solve our equation.
  2. Set your equation up in the form (x + _)(x + _) = 0. "Factoring" is just a term that means "finding the values that multiply together to give you something else." In this case, we're trying to break our quadratic equation down to its factors. Since the x term with the highest exponent is x2 (or, in other words, x × x), we'll start by setting up the factored form of the equation like this: (x + _)(x + _) = 0.
    • Note the blank spaces — in the next few steps, we'll fill these in to complete the factored equation.
  3. Find the factors of your "C" term. Next, list all of the numbers that can multiply together to give you the c term in your quadratic equation. These are its factors.
    • In our equation (x2 + 7x + 12 = 0), 12 is our c term. The numbers that can multiply to make 12 are: 1 and 12, 2 and 6, and 3 and 4. This means the factors of 12 are 1, 2, 3, 4, 6, and 12.
  4. Find the two factors of C that add up to your "B" term. From your short list of numbers that multiply together to make your c value, pick the two that add up to make your b term. To be clear, you're not looking for factors of your b term — just two numbers that add together to make it.
    • In our equation (x2 + 7x + 12 = 0), our b term is 7. Our list of factors for c are 1, 2, 3, 4, 6, and 12. 3 and 4 add up to 7, so these are the numbers we want.
    • If there weren't two numbers in this list of factors that added up to 7, we would say that the equation is "unfactorable over the integers."[3] Basically, for our purposes, this means that the equation can't be factored and that we'll have to use another method to find the roots.
  5. Fill in the spaces in your factored equation. Now, simply fill in the blanks in the factored form of the equation you've prepared with the two numbers you just chose from the list of factors. This gives you the factored form of your original quadratic equation.
    • Let's fill in the blanks in the factored form of our quadratic equation: (x + 3)(x + 4) = 0.
  6. Solve for both "x" values. Now, all you need to do to find the roots of your original quadratic equation is to set the terms in each parentheses equal to 0 and solve for x. Since the terms in parentheses are multiplied by each other, if either one equals zero, the entire equation will. Thus, the roots of the equations are the x values that get each set of terms in parentheses to equal zero.
    • In our example, our terms in parentheses are (x + 3) and (x + 4) = 0. Setting each of these equal to zero, we get:
    • x + 3 = 0: x = -3
    • x + 4 = 0: x = -4
    • Note that we can check these two answers exactly as we would if we had used the quadratic formula to find them.

Factoring With an "A" Value ≠ 1

  1. Break your "A" term down into its factors. If the a term in a quadratic equation isn't equal to one, it gets a little harder to factor, but it can still be possible. Start by breaking the a term into its factors — because there's an x2 in the a term, both factors will contain x.
    • In this section, let's use 2x2 + 14x + 12 = 0 as our example equation. In this case, 2x2 is our a term. Since 2 is a prime number, its only factors are 2 and 1. This means that, for our purposes, the factors of 2x2 are 2x and x.
    • Note that there are cases where there are more than two factors for the a term. If we were dealing with 8x2, for instance, we'd have 8x and x plus 2x and 4x. In this case, we've have to test both to find the set that fits.
  2. Set up your factored equation in the form ((factor 1)+ _)((factor 2) + _). We start to factor almost exactly as in the section above. However, this time, at least one of our x terms will have a coefficient next to it (sometimes, both will — it depends on the factors you've broken your a term into.)[3]
    • In our example, we would set up our equation like this: (2x + _)(x + _).
  3. Find the factors of your C term. This part is exactly the same as in the section above — you're just looking for the numbers that can multiply together to give you your c value.
    • In our example, since our c term is still 12, our list of factors is the same: 1, 2, 3, 4, 6, and 12.
  4. Plug in two numbers from your list that will give you your B term. This part is tricky — you want to pick two numbers that, when plugged in to the factored form of your equation, will give you the b term in your original quadratic equation. Keep in mind, though, that this time you won't just have two x's in the factored form of your equation — you'll have at least one x term with a coefficient.
    • In our example, our b term is 14x. This means we want to find two numbers from our list of c factors that, when we multiply one by 2x and the other by x, will give us a total of 14x.
    • Let's try our factors from before: 3 and 4. 3 × 2x = 6x, 4 × x = 4x. 4x + 6x = 10x. This doesn't work, so let's flip them around. 4 × 2x = 8x, 3 × x = 3x. 8x + 3x = 11x. We don't get 14x either way, so this set of factors isn't right.
    • Now, let's try 6 and 2. 6 × 2x = 12x, 2 × x = 2x. 12x + 2x = 14x. Success! We'll use 6 and 2 to fill in the blanks in our factored equation.
  5. Fill in the blanks and solve for x as normal. Now, use the two factors you just found to fill in the blanks in your factored equation. Keep in mind that you need to put each in the right space so that when they multiply by the x terms you get the correct b term. After this, just set each set of parentheses equal to zero and solve as before.
    • In this case, our equation would be (2x + 2)(x + 6) = 0. Setting each half equal to zero, we get:
    • 2x + 2 = 0
    • 2x = -2 : x = -1
    • x + 6 = 0 : x = -6



Tips

  • Remember that a square root can be both positive and negative. Don't fall into the trap of only writing down one answer when there should be two.
  • Note that, for some quadratic equations, there is an advanced method of solving known as "completing the square." See our article on the subject for a good step-by-step guide.
  • Believe it or not, factoring and completing the square are just two very roundabout ways of using the quadratic formula to solve equations. See our article on deriving the quadratic formula for a good breakdown, but be warned — things get complicated!

Related Articles

Sources and Citations

  1. ↑ http://education-portal.com/academy/lesson/what-is-a-quadratic-equation-definition-examples.html
  2. ↑ http://www.mathsisfun.com/quadratic-equation-solver.html
  3. ↑ 3.0 3.1 http://www.purplemath.com/modules/factquad2.htm
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