How to Find the Roots of a Quadratic Equation

Step-by-Step Guide

1. 1

   Step 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.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. 2

   Step 2: Plug a

   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) , 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. , 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. , 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., 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.
3. 3

   Step 3: and c into x = (-b +/-√(b2 - 4ac))/2a.

4. 4

   Step 4: Solve.

5. 5

   Step 5: Add and subtract to get two final answers.

6. 6

   Step 6: Check your answers.

7. 7

   Step 7: Alternatively

8. 8

   Step 8: check your answers manually.

Detailed Guide

About the Author