Algebra

Quadratic Discriminant

Quadratic Discriminant: Level 4 Challenges

         

What is the sum of all the integers aa such that the following equation has no real roots:

x2+x+a5x1=a?\frac{x^2+x+a-5}{x-1}=a?

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.

Find the sum of integral values of α\alpha such that 2log(x+3)=log(αx)2\log (x+3)=\log (\alpha x) has exactly one real solution and α<20|\alpha |<20.

If α\alpha is one of the non-real seventh roots of unity, then find the discriminant of the monic quadratic equation with the roots α+α2+α4\alpha+\alpha^2+\alpha^4 and α3+α5+α6.\alpha^3+\alpha^5+\alpha^6.

Details and Assumptions:

  • The discriminant of a quadratic equation ax2+bx+c=0ax^2+bx+c=0 is b24ac.b^2-4ac.

(a1)(x2+x+1)2(a+1)(x4+x2+1)=0(a-1)(x^{2}+x+1)^{2}-(a+1)(x^{4}+x^{2}+1)=0

Provided that two roots of the above equation are real and distinct for aRA,a \in \mathbb{R}-A, find the set A.A.

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