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.
To determine the values of k for which the function f(x) has two zeros, we need to find the discriminant of the quadratic equation formed by the given function. The discriminant is found using the formula: Δ = b^2 - 4ac, where a, b, and c are coefficients of the quadratic equation ax^2 + bx + c.
In this case, the quadratic equation is f(x) = 4x^2 - 3x + 2kx + 1. We can rewrite it as 4x^2 + (2k - 3)x + 1.
Comparing this equation with the general quadratic equation ax^2 + bx + c, we have:
a = 4
b = 2k - 3
c = 1
Now, let's calculate the discriminant: Δ = (2k - 3)^2 - 4(4)(1).
Expanding and simplifying this expression, we get:
Δ = 4k^2 - 12k + 9 - 16
= 4k^2 - 12k - 7
To find the values of k for which the function has two zeros, Δ must be greater than or equal to zero (Δ ≥ 0).
Therefore, we solve the inequality 4k^2 - 12k - 7 ≥ 0.
To do this, we can factor or use the quadratic formula. Let's use the quadratic formula:
k = (-b ± √Δ) / (2a)
Using the equation Δ = 4k^2 - 12k - 7 and substituting the values, we can calculate the solutions for k.
Once you find the values of k that satisfy the inequality, substitute those values into the original equation f(x) = 4x^2 - 3x + 2kx + 1 to check if it has two zeros.