Find the Zeros of a Function

The zero of a function is a value of x that makes the value of function equal to zero. You usually try to find the zeros of a function to find an "answer" to a polynomial equation, such as x2 + 4x +3 = 0. Follow these steps to learn several different ways how to find the zeros of a function.

Steps

Finding Zeros by Factoring

  1. Set up your equation so that it looks something like x2 + 5x + 4. Start with your biggest terms and move successively down, until you get to your constant. You want your big polynomial, such as or x2, to be the first term, and then to successively go down to smaller terms, until the last term is just a single number, like 8 or 4. Add an equals sign and a zero to the end of each term.
    • Polynomial terms that have been ordered correctly:
      • x2 + 5x + 6 = 0
      • x2 - 2x – 3 = 0
    • Polynomial terms that have been ordered incorrectly:
      • 5x + 6 = -x2
      • x2 = 2x + 3
  2. Give your equation an "a", "b", and "c" number. There's no math involved in this step. It's purely for setting things up so that you can factor easier when the time comes. Think of your equation as having a format. The format of the equation is ax2 ± bx ± c = 0. Simply find out the a, b, and c values in your equation. Here are some examples:
    • x2 + 5x + 6 = 0
      • a = 1 (no number in front of "x" = 1, as there is still one "x" being squared)
      • b = 5
      • c = 6
    • x2 - 2x – 3 = 0
      • a = 1 (no number in front of "x" = 1, as there is still one "x" being squared)
      • b = -2
      • c = -3
  3. Write down all the factor pairs of your "c" value. A factor pair is two numbers that, multiplied together, equal that number. Pay special attention to negative numbers. Two negative numbers multiplied together equal a positive number. Order doesn't matter here. ("1 x 4" is the same as "4 x 1".)
    • Equation: x2 + 5x + 6 = 0
    • Factor pairs of 6, or c:
      • 1 x 6 = 6
      • -1 x -6 = 6
      • 2 x 3 = 6
      • -2 x -3 = 6
  4. Find the factor pair that, when added together, equals "b". Look at your b value and find which of your factor pairs add up to that number.
    • b value = 5
    • Factor pair whose sum equals 5 = 2 and 3
      • 2 + 3 = 5
  5. Put that factor pair into two binomials. A binomial is just (x ± number)(x ± number). How do you know if it's a plus sign or minus sign that you put into the binomial? You look at the factor pair numbers: positive number = plus sign, negative number = minus sign. Here is the factor pair that we've put into a binomial:
    • (x + 2)(x + 3) = 0
  6. Solve each factor by moving the constant over to the other side of the equation. Break the two binomials apart — (x + 2) = 0 and (x + 3) = 0 — and then "solve" the equation by adding or subtracting to isolate the variable and the constant:
    • (x + 2) = 0 becomes x = -2
    • (x + 3) = 0 becomes x = -3
  7. Finished. These are the zeroes of your function.

Solve Quadratic Equations Using the Quadratic Formula

  1. Know the quadratic formula. The quadratic formula is as follows: the opposite of b plus or minus the square root of b^2-4ac, all over 2a.
  2. Give your equation an "a", "b", and "c" number. There's no math involved in this step. It's purely for setting things up so that you can factor easier when the time comes. Think of your equation as having a format. The format of the equation is ax2 ± bx ± c = 0. Simply find out the a, b, and c values in your equation.
  3. Having figured out your "a", "b", and "c" numbers, plug those numbers into the quadratic formula. You already know the numbers, and you have the quadratic formula right in front of you. Just plug in the a value whenever you see the a in the quadratic equation, and so on for the "bs" and "cs".
  4. Solve the equation. In order to solve the quadratic formula, you need to know how to divide, how to solve square roots, and how to work with fractions. Everything else is simple plug-and-chug.
    • Another variant of solving the quadratic equation is completing the square. Some people find this easier than solving a whole quadratic formula.
  5. Know that the two values created by your quadratic formula will be the "zeroes" you are looking for. Because of the square root will turn into a ± number (i.e. ±5), you'll have two different fractions. Both fractions, simplified, will be the answers to your function.

Finding Zeroes by Graphing

  1. Take your function and plug it into your graphing calculator. Your equation should be in the form of x2 + 8x + 12 = 0.
  2. Look for the two points where the graphed equation crosses the X-axis. These two points will be your zeroes, or answers to the function.
  3. Use this graphing technique more as a way to double-check your equation, not as a way to solve it. If you're graphing an equation in order to find its zeroes and you need to show work, use this method as a way of double-checking that the answers you got were actually correct. Most teachers will not assign credit for just the answers, without the work shown.



Tips

  • A zero is the same as the x- intercept.
  • You can double check your work by plugging your answers — one by one, not at the same time — back into your equation. If the equation equals zero, you've got the right answer.

Sources and Citations

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