Define the parabola y = a x 2 + b x + c such that a , b , care positive integers. Suppose that the roots of the quadratic equation a x 2 + b x + c = 0 are x 1 , x 2 such that | x 1 | , | x 2 | &gt; 1, then compute the minimum value of a b c and when the minimum occur what is the value of a + b + c.

Question

Define the parabola   y  =  a      x    2    +  b  x  +  c such that   a  ,  b  ,  care positive integers. Suppose that the roots of the quadratic equation   a      x    2    +  b  x  +  c  =  0 are       x    1    ,      x    2   such that       |        x    1        |    ,      |        x    2        |    &amp;gt;  1, then compute the minimum value of   a  b  c and when the minimum occur what is the value of   a  +  b  +  c.

Bridget Acevedo · Accepted Answer

f  (  x  )  =  a      x    2    +  b  x  +  c cannot have any positive root when   a,   b,   c are all positive. So we have       x    1    ,      x    2    &amp;lt;  −  1.The midpoint between the roots is   −  b      /    2  a. That must be less than   −  1, so we have                    (1)                                          −            b                                2            a                          &amp;lt;        −        1                ⟹                          b                      2            a                          &amp;gt;        1                ⟹                b        &amp;gt;        2        a            Furthermore the larger root is less than −1 too, so  −  1                    (2)                    f        (        −        1        )        &amp;gt;        0                ⟹                a        −        b        +        c        &amp;gt;        0                ⟹                c        &amp;gt;        b        −        a            Given these conditions, the smallest values   a,   b and   c can have are   a  =  1,   b  =  c  =  3. Unfortunately       x    2    +  3  x  +  3 doesn&#039;t have roots at all. Increasing   c can&#039;t help, so   b must be at least   4, and so   c  =  4 too.      x    2    +  4  x  +  4  has a double root at       x    1    =      x    2    =  −  2 and if that is allowed by the conditions we have   a  b  c  =  16 and   a  +  b  +  c  =  9.If   a was larger, say   a  =  2 conditions   (  1  ) and   (  2  ) would give   b  ≥  5 and   c  ≥  4, which would give   a  b  c  ≥  40, which is much larger so we can discount that.

Brianna Schmidt · Accepted Answer

Addition to expert&#039;s answer: If the double root is not allowed, we would need to go to higher   b still, so consider   b  =  5 - then we get   c  =  5 too. Now at last there are two different roots (we don&#039;t need to compute them, only observe that the discriminant is positive). The product abc is now   25 which is still less than   40, and the answer for   a  +  b  +  c is then   11.

Define the parabola y=ax^2+bx+c such that a,b,care positive integers. Suppose that the roots of the quadratic equation ax^2+bx+c=0 are x_1,x_2 such that |x_1|,|x_2|>1, then compute the minimum value of abc and when the minimum occur what is the value of a+b+c.

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