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CIE A Level Maths: Pure 3复习笔记8.1.4 Complex Roots of Polynomials
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Date: 2022年8月26日 上午10:55
Complex Roots of Quadratics
What are complex roots?
Complex numbers
provide solutions for quadratic equations which have no real roots
Complex roots occur when solving a quadratic with a negative discriminant
This leads to square rooting a negative number
How do we solve a quadratic equation with complex roots?
We solve an equation with complex roots in the same way we solve any other quadratic equations
If in the form
we can rearrange to solve
If in the form
we can complete the square or use the quadratic formula
We use the property
along with a manipulation of surds
When the coefficients of the quadratic equation are real, complex roots occur in
complex conjugate
pairs
If
is a root of a quadratic with real coefficients then
is also a root
When the coefficients of the quadratic equation are non-real, the solutions will not be complex conjugates
To solve these use the quadratic formula
How do we find a quadratic equation given a complex root?
We can find the equation of the form
if you are given a complex root in the form
We know that the complex conjugate
is another root,
This means that
and
are factors of the quadratic equation
Therefore
Writing this as
will speed up expanding
Expanding and simplifying gives us a quadratic equation where b and c are real numbers
Worked Example
Exam Tip
Once you have your final answers you can check your roots are correct by substituting your solutions back into the original equation.
You should get 0 if correct! [Note: 0 is equivalent to 0 + 0i]
Complex Roots of Cubics & Quartics
How many roots should a polynomial have?
How do we solve a cubic equation with complex roots?
How do we solve a quartic equation with complex roots?
How do we solve cubic/quartic equations with unknown coefficients?
Worked Example
转载自savemyexams
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