Question

The roots of the quadratic equation $x^{2}-2\sqrt{3}x-22 = 0 $ are :

• A. imaginary
• B. real, rational and equal
• C. real, irrational and unequal
• D. real, rational and unequal

Answer 1

The Correct Option is C

Given, :
\(x^{2}-2 \sqrt{3} x-22=0\)
\(D=b^{2}-4 a c=12+4 \times 22=100>0\)
 \(\therefore\) Roots are irrational, real and unequal.
So, the correct option is (C) : real, irrational and unequal.

Answer 2

Discriminant D :
D = b² - ac
Compare the given equation \(x^2-2\sqrt3-22=0\) with the standard equation i.e
\(ax^2+bx+c\) , so we get
⇒ \(a=1, b=-2\sqrt3,c=-22\)
By substituting these values into the discriminant, we obtain
\(⇒D=(-2\sqrt3)^2-4(1)(-22)\)
\(=12+88\)
\(=100\)
Now, nature of the root :
Since, D > 0 and is a perfect square. So the roots are real, rational and unequal.
So, the correct option is (C) : real, irrational and unequal.