a. one real root
b. two real roots
c. two imaginary roots
c. cannot be determined
a. one real root
b. two real roots
c. two imaginary roots
c. cannot be determined
This equation is x^2 - 4x - x + 4 = x^2 - 5x + 4 = 0. Factoring it yields (x - 1)(x - 4) = 0, so the roots are +1 and +4 which are both real. Another way to find the roots is to use the quadratic formula given by (-b + or - sqrt(b^2 - 4ac))/2a where a (the coefficient of the x^2 term) = 1, b (coefficient of the x term) = -5, and c (coefficient of the constant term) = 4. This gives (-(-5) + or - sqrt(5^2 - 4*1*4))/2*1 = (+5 + or - sqrt(25 - 16))/2 = (5 +- sqrt(9))/2 = (5 + or - 3)/2 = 8/2 (+4) or 2/2 (+1) as before. The answer is therefore b.
To identify the nature of roots, you have to analyze the discriminant (D)
ax^{2} + bx + c = 0
D = b^{2} - 4ac
There are two real roots, if D > 0
There is one real root, if D = 0
There are two imaginary roots, if D < 0
~~~~~~~~~~~
x^{2} - 4x - x + 4 = 0
x^{2} - 5x + 4 = 0
a = 1, b = - 5, c = 4
D = (-5)^{2} - 4 • 1 • 4
D = 9 > 0
Thus, the answer is
b. Two real roots