Question

Use the discriminant to describe the roots of each equation. Then select the best description. x2 - 6x + 12 = 0 double root real and rational root real and irrational root imaginary root

Answer 1

imaginary roots

Explanation:

For a quadratic in the form ...

`ax^2+bx+c=0`

the discriminant is ...

`d=b^2-4ac`

You have a=1, b=-6, c=12, so the discriminant is ...

d = (-6)² -4(1)(12) = 36 -48 = -12

When the discriminant is negative, both roots are complex. When the discriminant is not a perfect square, both roots are irrational. Here, the discriminant is negative and not a perfect square, so the roots are complex with an irrational imaginary part.

The best single descriptor is imaginary root.

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The roots are (-b±√d)/(2a) = (6 ± 2i√3)/2 = 3 ± i√3. These roots have a rational real part and an irrational imaginary part. When the number with an imaginary part has a non-zero real part, it is called "complex", rather than "imaginary."