Question

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

$5x^2 - 60x + 95 = 0$
a. real, equal, rational
b. real, unequal, rational
c. real, unequal, irrational
d. imaginary / complex

Explanation:

Step1: Recall discriminant formula

For quadratic equation \(ax^2 + bx + c = 0\), discriminant \(D = b^2 - 4ac\).
Here, \(a = 5\), \(b = -60\), \(c = 95\).

Step2: Calculate discriminant

\(D = (-60)^2 - 4\times5\times95\)
\(= 3600 - 1900\)
\(= 1700\)

Step3: Analyze discriminant

• \(D = 1700>0\), so roots are real and unequal.
• \(\sqrt{1700}=\sqrt{100\times17}=10\sqrt{17}\), which is irrational. So roots are real, unequal, irrational.

Answer:

C. Real, unequal, irrational