Quadratic Discriminant

Quadratic Discriminant - Problem Solving


If a,b,ca, b, c are real numbers such that c(a+b+c)<0 c (a+b+ c ) < 0 , what can we say about b24ac b^2 - 4ac ?

Hint: Think about the function f(x)=ax2+bx+c f(x) = ax^2 + bx + c .

For what values of mm are there no intersection points between the line y=mx+13 y= mx + 13 and the circle x2+y2=17 x^2 + y^2 = 17 ?

kk is uniformly chosen from the interval [5,5][ -5, 5] . Let pp be the probability that the quadratic f(x)=x2+kx+1 f(x) = x^2 + kx + 1 has both roots between -2 and 4. What is the value of 1000p \lfloor 1000 p \rfloor ?

Details and assumptions

Greatest Integer Function: x:RZ\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z} refers to the greatest integer less than or equal to xx. For example 2.3=2\lfloor 2.3 \rfloor = 2 and 3.4=4\lfloor -3.4 \rfloor = -4.

For how many positive integer values of c does the equation 2x2+705x+c=0 2x^2 + 705x +c = 0 have an integer solution?

For a positive integer mm such that m25,m \le 25, what is the probability that the quadratic equation x2+mx+1+m2=0x^2+mx+\frac{1+m}{2}=0 has real roots?


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