Question

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Answer 1

A quadratic always has two solutions. They could be two real number solutions (the parabola crosses the x-axis in two places), one real number double solution (the parabola just touches the x-axis in one spot) to two complex (imaginary) solutions where the parabola doesn't cross the x-axis.

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Answer 2

Answer:

If we consider a general quadratic equation:

ax2+bx+c=0

And suppose that we denote roots by α and β, then

x=α,β⇒(x−α)(x−β)=0

∴x2−(α+β)x+αβ=0

Equivalently we can write as

∴x2−(sum of roots)x+(product of roots)=0

And comparing these identical equations we can readily derive the following important relationships:

sum of roots=−ba and product of roots=ca

We also know that complex roots appear in conjugate pairs, so we can form some suitable equations.

Ex 1: α,β=1±2i

S=(1−2i)+(1+2i)=2

P=(1−2i)(1+2i)=1+4=5

So the equation with these roots is:

x2−2x+5=0

Ex 2: α,β=2±1i

S=(2−i)+

Step-by-step explanation:

Hope this helps!!