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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
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Systems of Equations
Vieta root jumping is a descent method that occurs when you have to solve a Diophantine equation whose solutions have a recursive structure. Popular in advanced math olympiad number theory problems.
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
If complex numbers \(x\) and \(y\) satisfy the simultaneous equations \[\begin{align} x^2+y^2 &= 4 \\ 2x^2-3y^2 &= -67, \end{align}\] what is the value of \((y+x)(y-x)\)?
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.
Let \(a\) and \(b\) be the values of non-zero real numbers \(x\) and \(y,\) respectively, that satisfy \[3x^2+18y-12x=-9, x^2+9y-7x=-6.\] What is the value of \(a^2+b^2?\)
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.
Let \(x=A\) and \(y=B\) be the solutions of the simultaneous equations \[\begin{align} \frac{3}{2x-1}+\frac{59}{y+1} &= 2 \\ \frac{3}{2x-1}-\frac{59}{y+1} &= 1. \end{align}\] What is the value of \(\frac{B}{A}\)?
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 \(x=a \; (< 0)\) and \(y=b\) are solutions of the simultaneous equations \[2x^2-2y^2+3x-56y=44,\, x^2-y^2+x-29y=22, \] what is the value of \(b-a\)?
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.
Let \(x=\alpha\), \(y=\beta\) and \(z=\gamma\) be the solutions of the simultaneous equations \[\begin{align} x+y-19xy &= 0 \\ y+z-12yz &= 0 \\ z+x-21zx &= 0, \end{align} \] where \( xyz \neq 0\). If the value of \(\alpha+\beta+\gamma\) can be expressed as \(\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers, what is \(a+b\)?