If the polynomial x^2+bx+c has exactly one real root and b=c+11, find the value of the product of all possible values of c.
+505 for the polynomial to have 1 real root, its discriminant, \(\sqrt{b^2-4ac}\), must be equal to zero.
In this case, it would mean that\(\sqrt{(c+11)^2-4c}\) must be equal to zero. We can then set up an equation:
\(\sqrt{(x+11)^2-4x}=0\\ (x+11)^2-4x=0\\ x^2+22x+121-4x=0\\ x^2+18x+121=0\)
(i replaced the variable c with x to make it less confusing)
the product of the roots of a quadratic is equal to the constant term divided by the coefficient of the quadratic term. In this case, it would be equal to 121/1 = 121.
Therefore, the answer to your question is c = 121.
