I need help with this question
determine the values in k for which the function f(x)= 4x^2-3x+2kx+1 has two zeros. Check these values in the original equation.
i don't get how to do it please help me.
F(x) = Y = 4x^2 - 3x + 2kx + 1.
This problem was solved by using EXCEL spread sheets and trial & error.
First, I temporarily ignored the 3rd term (2kx); and I changed b (the coefficient of x) until I found the required zeroes. Then I calculated the corresponding value of k:
b = -3 + 2k = 5,
-3 + 2k = 5,
2k = 5 + 3,
2k = 8,
k = 8 / 2 = 4.
5 was chosen for b because it gave 2
zeroes.
So when k = 4: (-1 , 0) , (-1/4 , 0).
To determine the values of k for which the function f(x) = 4x^2 - 3x + 2kx + 1 has two zeros, you need to find the values of k where the quadratic equation has discriminant equal to zero. The discriminant of a quadratic equation in the form ax^2 + bx + c = 0 is given by the formula:
D = b^2 - 4ac
For the given equation f(x) = 4x^2 - 3x + 2kx + 1, we have a = 4, b = -3, and c = 1. Substituting these values into the discriminant formula, we get:
D = (-3)^2 - 4(4)(1)
D = 9 - 16
D = -7
Since the discriminant D is negative (-7), it means that the quadratic equation has no real roots and, therefore, f(x) has no real zeros. This implies that there are no values of k for which the function f(x) = 4x^2 - 3x + 2kx + 1 has two zeros.
To confirm this, you can substitute any value of k into the original equation and solve for x. You will find that the resulting quadratic equation does not have any real solutions.