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The Quadratic Formula:
     The Discriminant and Graphs 
(page 3 of 3)

  • Solve x2 + 2x = 1. Round to two decimal places.

    I cannot apply the Quadratic Formula yet! The Formula only applies once I have "(quadratic) = 0", and I don't have that yet here. The first thing I have to do is move the 1 over, so I'll have "= 0" on the right-hand side:   Copyright Elizabeth Stapel 2000-2011 All Rights Reserved

      x2 + 2x 1 = 0

    Letting a = 1, b = 2, and c = 1, the Quadratic Formula gives me:

      x = -2.414, x = 0.414

    Then the answer is x = 2.41, x = 0.41, rounded to two decimal places.


Here's the graph:




 y = x^2 + 2x - 1

The x-intercepts (that is, the solutions from above) are marked in red.

This relationship between the value inside the square root (the discriminant), the type of solutions (two different solutions, one repeated solution, or no real solutions), and the number of x-intercepts (on the corresponding graph) of the quadratic is summarized in this table:

x2 2x 3 x2 6x + 9 x2 + 3x + 3
x = -1, 3 x = 3 x = -3/2  sqrt(3)i/2
a positive number
inside the square root
inside the square root
a negative number
inside the square root
two real solutions one (repeated) real solution two complex solutions
x^2 - 2x - 3 x^2 - 6x + 9 x^2 + 3x + 3
two distinct x-intercepts one (repeated) x-intercept no x-intercepts




Probably the most important thing to remember when using the Quadratic Formula (other than the Formula itself, which you should memorize) is that you must do each step clearly and completely, so you don't lose your denominators or plus-minuses or square roots. Don't skip stuff, and you should do fine.

Warning: If you get in the habit of "forgetting" the square root sign until the end when the back of the book "reminds" you that you "meant" to put it in, I'll bet good money that you'll mess up on your test. If you get in the habit of "forgetting" the plus/minus sign until the answer in the back "reminds" you that it belongs in there, then you will almost certainly miss every single problem where the answer doesn't have a square root symbol in it to "remind" you to put the plus/minus sign back in. That is, any time your answer is supposed to be something like "x = 5 10", you will put down "x = 5 + 10 = 15", and will have no idea how the book (or test) got the second answer of "x = 5". If you get sloppy with the denominator "2a", either by forgetting the "a" or by not dividing the entire numerator by this value, you will consistenly get the wrong answers.

I've been grading homework and tests for too many years to be kidding about this. Really, truly; you want to do your work neatly and completely every single time!

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Cite this article as:

Stapel, Elizabeth. "The Quadratic Formula: The Discriminant and Graphs." Purplemath.
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