Determine the Nature of Roots of the Quadratic Equation (3/5)x² − (2/3)x + 1 = 0
Solution
Given: $$\frac{3}{5}x^2-\frac{2}{3}x+1=0$$
Here, $$a=\frac{3}{5},\quad b=-\frac{2}{3},\quad c=1$$
Using the discriminant, $$D=b^2-4ac$$
$$D=\left(-\frac{2}{3}\right)^2-4\left(\frac{3}{5}\right)(1)$$
$$D=\frac{4}{9}-\frac{12}{5}$$
$$D=\frac{20-108}{45}=-\frac{88}{45}$$
Since $$D<0,$$ the roots are imaginary (non-real) and distinct.
Answer
The equation (3/5)x² − (2/3)x + 1 = 0 has two distinct imaginary (non-real) roots.