Explanatory Answer:
The given equation is in the form of a quadratic equation, where the variable is n. We can rewrite the equation as follows:
5n^2 − n − 13 = 0
To find the discriminant of this equation, we can use the formula:
discriminant = b^2 - 4ac
where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. In our case, a = 5, b = -1, and c = -13. So, the discriminant is:
b^2 - 4ac = (-1)^2 - 4(5)(-13) = 1 + 260 = 261
Therefore, the discriminant of the equation is 261.
To determine the number of solutions of the equation, we can use the following rules:
If the discriminant is positive, the equation has two real solutions.
If the discriminant is zero, the equation has one real solution.
If the discriminant is negative, the equation has no real solutions.
Since the discriminant in our case is positive (261 > 0), the equation has two real solutions.
To solve the equation using the quadratic formula, we can use the following formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
where x is the solution of the equation. Plugging in the values of a, b, and c, we get:
n = (-(-1) ± sqrt((-1)^2 - 4(5)(-13))) / 2(5)
n = (1 ± sqrt(1 + 260)) / 10
n = (1 ± sqrt(261)) / 10
Therefore, the two solutions of the equation are:
n = (1 + sqrt(261)) / 10 ≈ 1.601 or n = (1 - sqrt(261)) / 10 ≈ -0.807
So, the solutions of the equation are approximately n = 1.601 and n = -0.807.