We all know that the discriminant is the part b 2 − 4 a c of the equation x = − b ± b 2 − 4 a c 2 a that we use to find the roots of a quadratic equation eg: a x 2 + b x + c = 0 or the part in a trinomial expression like a x 2 + b x + c .What I want to know is what kind of ideas we can conclude for a given inequality of a discriminant.

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We all know that the discriminant is the part       b    2    −  4  a  c of the equation  x  =            −      b      ±                        b          2                −        4        a        c                    2      a      that we use to find the roots of a quadratic equation eg:   a      x    2    +  b  x  +  c  =  0 or the part in a trinomial expression like   a      x    2    +  b  x  +  c .What I want to know is what kind of ideas we can conclude for a given inequality of a discriminant.

boomzwamhc · Accepted Answer

Step 1Let me rewrite the quadratic equation as  x  =  −      b          2      a        ±      −          1      a              (                        4          a          c          −                      b                          2                                                4          a                    )        . .Notice that I wrote it in terms of the coordinates of the vertex. Now, let us consider some cases about the vertex (h, k), assuming that the quadratic opens upward.1)   k  &amp;lt;  0 .In this case, the vertex is below the x-axis and the quadratic must have two real roots. Now,                                          4            a            c            −                          b                              2                                                          4            a                          &amp;lt;        0                            4        a        c        −                  b                      2                          &amp;lt;        0                            −        (        4        a        c        −                  b                      2                          )        &amp;gt;        0                                      b                      2                          −        4        a        c        &amp;gt;        0            which matches our criteria that a quadratic has two real roots if and only if       b          2        −  4  a  c  &amp;gt;  0 .2)   k  =  0 .This means that the vertex is on the x-axis and the quadratic has a double root. Now, making a similar solution from the first case,                                          4            a            c            −                          b                              2                                                          4                          a                              2                                                                        =        0                            4        a        c        −                  b                      2                                              =        0                            −        4        a        c        +                  b                      2                                              =        0                                      b                      2                          −        4        a        c                            =        0            This matches our criteria that a quadratic has a double root if and only if       b          2        −  4  a  c  =  0 .3)   k  &amp;gt;  0 .In this case, the vertex is above the x-axis and since it is the minimum of the function, the quadratic does not have any real root. Now,                                          4            a            c            −                          b                              2                                                          4            a                          &amp;gt;        0                            4        a        c        −                  b                      2                          &amp;gt;        0                            −        4        a        c        +                  b                      2                          &amp;lt;        0                                      b                      2                          −        4        a        c        &amp;lt;        0            and this matches our criteria that a quadratic does not have any real root if and only if       b          2        −  4  a  c  &amp;lt;  0 .I&#039;ll leave the case where a quadratic opens downward to you.

We all know that the discriminant is the part b 2 </msup> −

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