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The Quadratic Formula:
     Solutions and the Discriminant
(page 2 of 3)


  • Solve x(x 2) = 4. Round your answer to two decimal places.

    I not only cannot apply the Quadratic Formula at this point, I cannot factor either.

    I can not claim that "x = 4, x 2 = 4", because this is not how "solving by factoring" works. I must first rearrange the equation in the form "(quadratic) = 0", whether I'm factoring or using the Quadratic Formula. The first thing I have to do here is multiply through on the left-hand side, and then I'll move the 4 over:

      x(x 2) = 4
      x2 2x = 4
      x2 2x 4 = 0

    Since there are no factors of (1)(4) = 4 that add up to 2, then this quadratic does not factor. (In other words, there is no possible way that the faux-factoring solution of "x = 4, x 2 = 4" could be even slightly correct.) So factoring won't work, but I can use the Quadratic Formula; in this case, a = 1, b = 2, and c = 4:   Copyright Elizabeth Stapel 2000-2011 All Rights Reserved

      x = -1.24, x = 3.24

    Then the answer is:  x = 1.24, x = 3.24, rounded to two places.

  

   

For reference, here's what the graph looks like:

  

y = x^2 - 2x - 4
   


There is a connection between the solutions from the Quadratic Formula and the graph of the parabola: you can tell how many x-intercepts you're going to have from the value inside the square root. The argument of the square root, the expression b2 4ac, is called the "discriminant" because, by using its value, you can discriminate between (tell the differences between) the various solution types.

  • Solve 9x2 + 12x + 4 = 0.

    Using a = 9, b = 12, and c = 4, the Quadratic Formula gives:

      x = -2/3

    Then the answer is x = 2/3

In the previous examples, I had gotten two solutions because of the "plus-minus" part of the Formula. In this case, though, the square root reduced to zero, so the plus-minus didn't count for anything.

This solution is called a "repeated" root, because x is equal to 2/3, but it's equal kind of twice: 2/3 + 0 and 2/3 0. You can also see this repetition better if you factor:  9x2 + 12x + 4 = (3x + 2)(3x + 2) = 0, so x = 2/3 and x = 2/3. Any time you get zero in the square root of the Quadratic Formula, you'll only get one solution.

The square-root part of the Quadratic Formula is called "the discriminant", I suppose because you can use it to discriminate between whether the given quadratic has two solutions, one solution, or no solutions.

  

This is what the graph looks like:

  

y = 9x^2 + 12x + 4

The parabola only just touches the x-axis at x = 2/3; it doesn't actually cross. This is always true: if you have a root that appears exactly twice, then the graph will "kiss" the axis there, but not pass through.

  • Solve 3x2 + 4x + 2 = 0.

    Since there are no factors of (3)(2) = 6 that add up to 4, this quadratic does not factor. But the Quadratic Formula always works; in this case, a = 3, b = 4, and c = 2:

      x = [-4+/-sqrt(-8)]/6

    At this point, I have a negative number inside the square root. If you haven't learned about complex numbers yet, then you would have to stop here, and the answer would be "no solution"; if you do know about complex numbers, then you can continue the calculations:

      x = -2/3 +/- sqrt(2)i/3

If you do not know about complexes, then your answer would be "no solution". If you do know about complexes, then you would say there there is a "complex solution" and would give the answer (shown above) with the " i " in it. But whether or not you know about complexes, you know that you cannot graph your answer, because you cannot graph the square root of a negative number. There are no such values on the x-axis. Since you can't find a graphable solution to the quadratic, then reasonably there should not be any x-intercept (because you can graph an x-intercept).

  

Here's the graph:

  

 y = 3x^2 + 4x + 2 = 0

This relationship is always true: If you get a negative value inside the square root, then there will be no real number solution, and therefore no x-intercepts. (The relationship between the value inside the square root, the type of solutions, and the number of x-intercepts on the graph is summarized in a chart on the next page.)

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Cite this article as:

Stapel, Elizabeth. "The Quadratic Formula: Solutions and the Discriminant." Purplemath.
    Available from 
http://www.purplemath.com/modules/quadform2.htm.
    Accessed
 

 



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