Question

1. for the following quadratic equation, determine the nature of the roots.

$x^2 - 12x + 21 = 0$
a. real, equal, rational
b. real, unequal, rational
c. real, unequal, irrational
d. imaginary / complex

Explanation:

Step1: Recall discriminant formula

For a quadratic equation \(ax^2 + bx + c = 0\), the discriminant \(D = b^2 - 4ac\).
Here, \(a = 1\), \(b = -12\), \(c = 21\).

Step2: Calculate discriminant

\(D = (-12)^2 - 4\times1\times21\)
\(= 144 - 84\)
\(= 60\)

Step3: Analyze discriminant

• If \(D > 0\), roots are real and unequal.
• If \(D = 0\), roots are real and equal.
• If \(D < 0\), roots are imaginary/complex.

Also, if \(D\) is a perfect square, roots are rational; else, irrational.
Since \(D = 60>0\) (so real, unequal) and \(60\) is not a perfect square (so irrational).

Answer:

C. Real, unequal, irrational