Final answer:
To solve the quadratic equation 5x² - x - k = 0 with a root difference of 1.8, we use the quadratic formula and set the discriminant part to 1.8 times the absolute value of a, and solve for k.
Step-by-step explanation:
The student is dealing with a quadratic equation 5x² - x - k = 0, where the difference between its roots is 1.8. To find the roots, we can apply the quadratic formula, which is derived from any equation of the form ax² + bx + c = 0. The formula for finding the roots is x = √((-b ± √(b² - 4ac)) / (2a)). The difference between the roots of a quadratic equation ax² + bx + c = 0 is given by √(b² - 4ac) / |a|. So, for the given equation 5x² - x - k = 0, the difference is 1.8, which means √(1² - 4(5)(-k)) / |5| = 1.8. We can solve this equation for k to find the specific value and then substitute it back into the quadratic formula to get the exact roots.