Determine the values of k for which the function f(x)= 4x^2-3x+2kx+1 has two zeros , check these values in the original equation
to have two zeros, either real or complex,
b^2 - 4ac ≠ 0
4x^2 + x(2k - 3) + 1
discrim. = (2k-3)^2 - 4(4)(1)
= 4k^2 - 12k + 9 - 16
= 4k^2 - 12k - 7
to have only 1 root
4k^2 - 12k - 7 = 0
(2k - 7)(2k + 1) = 0
k = 7/2 or k = -1
So to have two roots, k ≠ 7/2 or k ≠ -1
So put any value of k other than those two and you will get 2 differentroots.
Well, well, well, let's find those special values of k, shall we?
To find the values of k for which the quadratic function f(x) = 4x^2-3x+2kx+1 has two zeros, we need to use a little trick. You see, a quadratic function has two zeros when its discriminant, b^2-4ac, is greater than zero. In this case, a is 4, b is (2k-3), and c is 1.
So, let's plug these values into the discriminant formula:
(2k-3)^2 - 4 * 4 * 1 > 0
Expanding and simplifying:
4k^2 - 12k + 9 - 16 > 0
4k^2 - 12k - 7 > 0
Now, let's solve this inequality like a true mathematician. The best way to solve quadratic inequalities is by factoring (if possible), so let's give it a shot:
(2k - 7)(2k + 1) > 0
Now, we need to find the intervals where the expression on the left side is greater than zero. This means that either both factors are positive or both are negative.
Case 1: Both factors are positive
2k - 7 > 0 and 2k + 1 > 0
Solving each inequality separately:
2k > 7 --> k > 3.5
2k > -1 --> k > -0.5
Since we want both factors to be positive, the only valid solution is k > 3.5.
Case 2: Both factors are negative
2k - 7 < 0 and 2k + 1 < 0
Again, solving each inequality separately:
2k < 7 --> k < 3.5
2k < -1 --> k < -0.5
Here, we want both factors to be negative, so the only valid solution is k < -0.5.
In conclusion, for the quadratic function f(x) = 4x^2-3x+2kx+1 to have two zeros, k must be either greater than 3.5 or less than -0.5.
As for checking these values in the original equation, that's a piece of cake. Simply substitute each value of k into f(x) and see if you get two distinct zeros. Go ahead and give it a try, my friend!
To determine the values of k for which the given function has two zeros, we need to find the discriminant and set it greater than zero.
The discriminant (D) of a quadratic equation in the form ax^2 + bx + c = 0 is given by D = b^2 - 4ac.
In this case, the quadratic equation is f(x) = 4x^2 - 3x + 2kx + 1, which can be rewritten as 4x^2 + (2k - 3)x + 1 = 0.
Comparing with the general form ax^2 + bx + c = 0, we have:
a = 4
b = 2k - 3
c = 1
The discriminant can be calculated as:
D = (2k - 3)^2 - (4)(4)(1)
D = 4k^2 - 12k + 9 - 16
D = 4k^2 - 12k - 7
For the given equation to have two zeros, the discriminant D must be greater than zero. Therefore, we can set up the inequality:
D > 0
4k^2 - 12k - 7 > 0
To solve this inequality, we can factorize or apply the quadratic formula. However, since the quadratic equation is not easily factorizable, we will use the quadratic formula.
Using the quadratic formula:
k = (-b ± √(b^2 - 4ac)) / (2a)
k = (-(-12) ± √((-12)^2 - 4(4)(-7))) / (2(4))
k = (12 ± √(144 + 112)) / 8
k = (12 ± √256) / 8
k = (12 ± 16) / 8
Therefore, we get two possible values for k:
1. k = (12 + 16) / 8 = 28 / 8 = 7/2
2. k = (12 - 16) / 8 = -4 / 8 = -1/2
To check these values in the original equation, we substitute them back into f(x) and verify if they result in zero:
1. For k = 7/2:
f(x) = 4x^2 - 3x + 2(7/2)x + 1
= 4x^2 - 3x + 7x + 1
= 4x^2 + 4x + 1
This equation has two zeros since it is simply a perfect square trinomial, which factors as:
f(x) = (2x + 1)^2
Therefore, k = 7/2 is a valid value.
2. For k = -1/2:
f(x) = 4x^2 - 3x + 2(-1/2)x + 1
= 4x^2 - 3x - x + 1
= 4x^2 - 4x + 1
This equation also has two zeros since it is another perfect square trinomial, which factors as:
f(x) = (2x - 1)^2
Therefore, k = -1/2 is also a valid value.
In summary, the values of k for which the function f(x) = 4x^2 - 3x + 2kx + 1 has two zeros are k = 7/2 and k = -1/2.
To determine the values of k for which the function f(x) = 4x^2 - 3x + 2kx + 1 has two zeros, we need to find the discriminant of the quadratic equation.
The quadratic equation is given by ax^2 + bx + c = 0, where a = 4, b = (-3 + 2k), and c = 1.
The discriminant (D) is given by D = b^2 - 4ac.
In our equation, the discriminant is:
D = (-3 + 2k)^2 - 4(4)(1)
Simplifying the expression, we get:
D = 9 - 12k + 4k^2 - 16
D = 4k^2 - 12k - 7
For the function to have two zeros, the discriminant (D) must be greater than zero, D > 0.
So, we have:
4k^2 - 12k - 7 > 0
To solve this inequality, we can use factoring or the quadratic formula.
Using the quadratic formula, the solutions for the inequality are:
k > (12 + √232)/8 or k < (12 - √232)/8
Simplifying further, we get:
k > (6 + √58)/4 or k < (6 - √58)/4
These are the values of k for which the function f(x) = 4x^2 - 3x + 2kx + 1 has two zeros.
To check these values in the original equation, substitute them into f(x) and see if the equation equals zero.
For example, for the value k = (6 + √58)/4:
f(x) = 4x^2 - 3x + 2((6 + √58)/4)x + 1
Simplify and solve the equation:
0 = 4x^2 - 3x + (3/2)x√58 + 1
Check if the equation equals zero for different values of x, and repeat the process for the other value of k as well.