For what values of k will the function f(x)=3x^2+4x+k=0 have no zeros, one zero, 2 zeros
Well, let me put on my clown shoes and entertain you with some mathematical humor!
For the function f(x) = 3x^2 + 4x + k = 0:
- If the discriminant (b^2 - 4ac) is negative, there will be no real zeros. So, for no zeros, we need to have: 4^2 - 4(3)(k) < 0. Simplifying, we get k > 8/3.
- If the discriminant is zero, there will be one real zero. Therefore, for one zero, we need: 4^2 - 4(3)(k) = 0. Solving this equation, we find k = 4/3.
- If the discriminant is positive, there will be two real zeros. Thus, for two zeros, we need: 4^2 - 4(3)(k) > 0. Simplifying, we get k < 8/3.
So, in summary:
- For no zeros: k > 8/3
- For one zero: k = 4/3
- For two zeros: k < 8/3
And that's how clown logic tries to entertain while answering math questions!
To determine the number of zeros for the quadratic function f(x) = 3x^2 + 4x + k = 0, we can use the discriminant, which is given by b^2 - 4ac.
If the discriminant is positive (b^2 - 4ac > 0), then the quadratic equation will have two distinct real zeros.
If the discriminant is zero (b^2 - 4ac = 0), then the quadratic equation will have one real zero (a double root).
If the discriminant is negative (b^2 - 4ac < 0), then the quadratic equation will have no real zeros.
For our function f(x) = 3x^2 + 4x + k, we can compare it to the standard form of a quadratic equation: ax^2 + bx + c = 0.
From the equation, we have a = 3, b = 4, and c = k.
Now, we can calculate the discriminant b^2 - 4ac:
Discriminant = 4^2 - 4 * 3 * k
= 16 - 12k
To determine the number of zeros, we need to look at the value of the discriminant.
1. If the discriminant is positive (16 - 12k > 0), the quadratic equation will have two distinct real zeros. Therefore, we need to solve the inequality:
16 - 12k > 0
Solving for k:
16 > 12k
k < 16/12
k < 4/3
So, for k < 4/3, the quadratic equation will have two zeros.
2. If the discriminant is zero (16 - 12k = 0), the quadratic equation will have one real zero (a double root). Therefore, we need to solve the equation:
16 - 12k = 0
Solving for k:
12k = 16
k = 16/12
k = 4/3
So, when k = 4/3, the quadratic equation will have only one zero.
3. If the discriminant is negative (16 - 12k < 0), the quadratic equation will have no real zeros. Therefore, we need to solve the inequality:
16 - 12k < 0
Solving for k:
16 < 12k
k > 16/12
k > 4/3
So, for k > 4/3, the quadratic equation will have no zeros.
In summary:
- For k < 4/3, the function f(x) = 3x^2 + 4x + k = 0 will have two zeros.
- When k = 4/3, the function f(x) = 3x^2 + 4x + k = 0 will have one zero.
- For k > 4/3, the function f(x) = 3x^2 + 4x + k = 0 will have no zeros.
To determine the number of zeros of the quadratic function, we can consider the discriminant of the equation. The discriminant is given by the expression b^2 - 4ac, where a, b, and c are the coefficients of the quadratic function f(x) = ax^2 + bx + c.
For the given function f(x) = 3x^2 + 4x + k, the coefficients are a = 3, b = 4, and c = k. So the discriminant is:
D = b^2 - 4ac
D = (4)^2 - 4(3)(k)
D = 16 - 12k
We can now analyze the discriminant to determine the number of zeros:
1. If the discriminant is greater than zero (D > 0), the function will have two distinct real zeros.
2. If the discriminant is equal to zero (D = 0), the function will have one real zero (repeated).
3. If the discriminant is less than zero (D < 0), the function will have no real zeros.
Now, let's analyze each case:
1. Two real zeros (D > 0):
For the function to have two distinct real zeros, the discriminant must be greater than zero:
16 - 12k > 0
To solve this inequality, we can isolate k:
16 > 12k
k < 16/12
k < 4/3
Therefore, the values of k that will result in the function having two distinct real zeros are k < 4/3.
2. One real zero (D = 0):
For the function to have one real zero (repeated), the discriminant must be equal to zero:
16 - 12k = 0
To solve this equation, we can isolate k:
12k = 16
k = 16/12
k = 4/3
Therefore, the value of k that will result in the function having one real zero is k = 4/3.
3. No real zeros (D < 0):
For the function to have no real zeros, the discriminant must be less than zero:
16 - 12k < 0
To solve this inequality, we can isolate k:
12k > 16
k > 16/12
k > 4/3
Therefore, the values of k that will result in the function having no real zeros are k > 4/3.
In summary:
- For k < 4/3, the quadratic function will have two distinct real zeros.
- For k = 4/3, the quadratic function will have one real zero (repeated).
- For k > 4/3, the quadratic function will have no real zeros.
you will have to look at the discriminant b^2 - 4ac
b^2-4ac = 16 - 4(3)(k) = 16 - 12k
If 16-12k = 0 you have one zero
if 16 - 12k > 0 you have 2 zeros
if 16-12k < 0 you have no zeros
solve each case to find the corresponding value of k