MHB Richard Perito: Find Quadratic with Roots -1 +/- i√2

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To find a quadratic equation with roots -1 + i√2 and -1 - i√2, the roots can be expressed as x = -1 ± i√2. Using the quadratic formula, the coefficients are identified as a = 1, b = 2, and c = 3, derived from the discriminant calculation. The resulting quadratic equation is f(x) = x² + 2x + 3, which has the specified roots. This equation confirms the relationship between the roots and the standard form of a quadratic. The discussion effectively demonstrates the process of deriving the quadratic from its roots.
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Here is the question:

Write a quadratic equation with the solution set -1 + i√2, -1 - i√2?

I have posted a link there to this topic so the OP can see my work.
 
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Hello Richard Perito,

I would begin by writing the roots as:

$$x=-1\pm i\sqrt{2}=\frac{-2\pm\sqrt{-8}}{2}$$

And so from the quadratic formula:

$$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

we see we have:

$$a=1,\,b=2,\,b^2-4ac=4-4c=-8\implies c=3$$

Hence, the quadratic:

$$f(x)=x^2+2x+3$$

has the given roots.
 
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