Waste less time on Facebook — follow Brilliant.
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Vieta's Formula
Vieta's formula relates the coefficients of polynomials to the sum and products of their roots. This can provide a shortcut to finding solutions in more complicated algebraic polynomials.
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Concept Quizzes
\(\alpha\), \(\beta\), and \(\gamma\) are the roots of the cubic equation \(x^3-x^2+2x-13=0\). If \(\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}=\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, what is the value of \(a+b\)?
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Given that 2 roots of \(f(x) = x^3 + ax + b\) are 2 and 7, what is \( a+b \)?
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
What is the product of all roots to the equation
\[ \begin{align} & (x-1)(x-2)(x-3) + (x-2)(x-3)(x-4) \\ + & (x-3)(x-4)(x-5) + (x-4)(x-5)(x-6) \\ + & (x-5)(x-6)(x-7) + (x-6)(x-7)(x-8) =0 ? \end{align} \]
Details and assumptions
Clarification: Make sure you scroll to the right (if need be) to see the full equation. This problem ends with a "?".
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
If the quartic \( x^4 + 3x^3 + 11 x^2 + 15 x + A \) has roots \( k, l, m\) and \(n \) such that \(kl = mn \), determine \(A\).
Excel in math and science
Master concepts by solving fun, challenging problems.
It's hard to learn from lectures and videos
Learn more effectively through short, conceptual quizzes.
Our wiki is made for math and science
Master advanced concepts through explanations,
examples, and problems from the community.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
If the cubic equation \(x^3-3x^2+5=0\) has three roots \(\alpha\), \(\beta\) and \(\gamma\), what is the value of \[(\alpha+\beta)(\beta+\gamma)(\gamma+\alpha)?\]