In order to determine the number of zeros the function f(x) = 3x^2 - 4x + k has, we can rely on the discriminant of the quadratic equation.
The discriminant (Δ) is given by Δ = b^2 - 4ac, where the quadratic equation is ax^2 + bx + c = 0.
For f(x) = 3x^2 - 4x + k, the coefficients are a = 3, b = -4, and c = k.
If Δ > 0, the quadratic equation has two distinct real roots.
If Δ = 0, the quadratic equation has one real root (a repeated root).
If Δ < 0, the quadratic equation has no real roots.
Substituting these values into the discriminant formula, we have:
Δ = (-4)^2 - 4(3)(k)
Δ = 16 - 12k
Now, let's analyze the possible values of k for each case:
1. One zero:
For the quadratic equation to have one zero, we need Δ = 0. Therefore:
16 - 12k = 0
12k = 16
k = 16/12
k = 4/3
2. Two zeros:
For the quadratic equation to have two zeros, we need Δ > 0. Therefore:
16 - 12k > 0
12k < 16
k < 16/12
k < 4/3
3. No zeros:
For the quadratic equation to have no zeros, we need Δ < 0. Therefore:
16 - 12k < 0
12k > 16
k > 16/12
k > 4/3
In summary:
- If k = 4/3, f(x) = 3x^2 - 4x + (4/3) will have one zero.
- If k < 4/3, f(x) = 3x^2 - 4x + k will have two zeros.
- If k > 4/3, f(x) = 3x^2 - 4x + k will have no zeros.
for what values of k will the function f(x)=3x^2-4x+k have one zero, one, two?
1 answer