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Quadratic Discriminant

This one number can tell you whether the solutions to a quadratic equation are real or non-real, and whether they are distinct or repeated.

Level 4

         

What is the sum of all the integers \(a\) such that the following equation has no real roots:

\[\frac{x^2+x+a-5}{x-1}=a?\]

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

Details and assumptions

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

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

If \(\alpha\) is one of the non-real seventh roots of unity, then find the discriminant of the monic quadratic equation with the roots \(\alpha+\alpha^2+\alpha^4\) and \(\alpha^3+\alpha^5+\alpha^6\).

\[\] Details and assumptions
The discriminant of a quadratic equation \(ax^2+bx+c=0\) is \(b^2-4ac.\)

\[(a-1)(x^{2}+x+1)^{2}-(a+1)(x^{4}+x^{2}+1)=0\] Provided that two roots of the above equation are real and distinct for \(a \in \mathbb{R}-A\). then find the set \(A\).

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