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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
Master advanced concepts through explanations,
examples, and problems from the community.
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including olympiad champions, researchers, and professionals.
Completing The Square
Learn how to eliminate the linear term and see through messy quadratics. You’ll know just how high to shoot your three-pointer to avoid tall defenders.
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
Consider two functions \[\begin{align} f(x) &= -4x^2 + 14 x + 4 \\ g(x) &= -7 x^2 + 8 x + 3. \end{align}\] Let \( a \) be the maximum value of \( f(x) + g(x) ,\) and \(b\) the value of \(x \) at that moment. Then 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.
For real numbers \(x\) and \(y\), the polynomial \[x^2+2y^2+2xy-8x-10y+30\] has a minimum value \(m\) at \(x=a\) and \(y=b\). What is the value of \(abm\)?
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 \(f(x)= 5 x + 60\) and \(g(x)= 4 x + 80.\) If \(a\) is the minimum value of \( f(x) \times g(x) \) and \(b\) is the value of \(x \) at that moment, 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.
Consider two points \(A=(2,5)\) and \(B=(4,4)\), and a third point \(P=(a, 0)\) on the \(x\)-axis. If the minimum value of \(\overline{PA}^2+\overline{PB}^2\) is \(c\), what is \(a+c\)?
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.
Consider the parabola \[y = x^2 + k^4 + p^4 -2k^2 x -2p^2x + 2k^2p^2 + 36,\] where \(k\) and \(p\) are parameters. If the vertex of the parabola lies on the line \[ y = \frac{ 1 }{25} x, \] what is the value of \( k^2 + p^2? \)