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Excel in math and science
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Algebra
Complex Numbers
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
Let \( z = a^2 - 5a - 36 + (a^2 + 6a + 8)i \). What is the sum of all real numbers \(a\) such that \( z^2 \) is a negative real number?
Details and assumptions
\(i\) is the imaginary number that satisfies \(i^2 = -1\).
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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by Brilliant Staff
If the quadratic equation \[ (2 + i)x^2 + (a - i)x + 46 - 2i = 0 \] has real roots, what is the value of the positive real constant \(a\)?
Details and assumptions
\(i\) is the imaginary number that satisfies \(i^2 = -1\).
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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by Brilliant Staff
Let \(z=x(1-i)+8(-2+i),\) where \(x\) is a real number. If \(z^2\) is a negative real number, what is the value of \(x?\)
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.
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by Brilliant Staff
Real numbers \(x\) and \(y\) satisfy the equation: \[ \frac{x}{1+i} + \frac{y}{1-i} = \frac{148}{12 + 2i}. \] What is the value of \(xy?\)
Details and assumptions
\(i\) is the imaginary number satisfying \(i^2 = -1\).
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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by Brilliant Staff
\( \frac{12 - \sqrt{-9}}{12 + \sqrt{-9}} + \frac{12 + \sqrt{-9}}{12 - \sqrt{-9}} \) can be simplified to \( \frac{a}{b} \), 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.
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That seems reasonable.
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by Brilliant Staff