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Excel in math and science
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Algebra
Completing The Square
Excel in math and science
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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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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,
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by Brilliant Staff
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,
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by Brilliant Staff
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
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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
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Our wiki is made for math and science
Master advanced concepts through explanations,
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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 \( k^2 + p^2? \)
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
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including olympiad champions, researchers, and professionals.
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by Brilliant Staff