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Excel in math and science
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Complex Numbers
"Complex" numbers share many properties with the real numbers. In fact, complex numbers include all the real numbers, so you know lots of them already. Such an overachiever.
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
A quadratic polynomial with real coefficients has \( 19 + 8 i \) as a root. If the other root has the form \( a + bi \), where \(a\) and \(b\) are real numbers, 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.
If \[ x (x-4-9i)(x-a+bi) = x^3 -8x^2+97x, \] what is \( a + b \) ?
Details and assumptions
\( i \) is the the imaginary unit, defined by \( i^2 = -1 \).
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.
Used and loved by 4 million people
Learn from a vibrant community of students and enthusiasts,
including olympiad champions, researchers, and professionals.
Suppose that \[7x^2-14x+182=c(x-a-bi)(x-a+bi),\] where \(i\) is the imaginary number that satisfies \(i^2=-1\) and \( b > 0. \) What is the value of \(a+b+c\)?
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.
Suppose that \[\begin{align} & x^4+6x^3+17x^2-6x-18 \\ &= (x+a)(x-a)(x+b+ci)(x+b-ci), \end{align} \] where \(i\) is the imaginary number that satisfies \(i^2=-1\). What is the value of \(a+b+c\)?
Details and assumptions
Assume \( a > 0 \) and \( c > 0 \).
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.
It is given that the cubic polynomial \( f(x) \) has real coefficients, and \(f(x) = 0 \) has a complex root \( 28 - 15 i \). If the sum of all the complex, non-real roots of \( f(x) = 0 \) can be written as \( a + bi \), what is the value of \( a + b\)?