For what values of k will the functon f(x)=kx^2-4x+k have no zeros

Determine the values of k for which the function f(x)=4x^2-3x+2kx+1 have 2 zeros

Well, for the function f(x) = kx^2 - 4x + k to have no zeros, the discriminant should be negative. The discriminant is given by b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation. In this case, the discriminant is:

(-4)^2 - 4(k)(k) = 16 - 4k^2

To have no zeros, the discriminant should be negative. So, we can set up the inequality:

16 - 4k^2 < 0

Simplifying this inequality, we get:

k^2 > 4

Taking the square root of both sides (remembering to include the negative value), we'll have:

k > 2 or k < -2

Therefore, the values of k for which the function f(x) = kx^2 - 4x + k has no zeros are k > 2 or k < -2.

As for the second question, for the quadratic function f(x) = 4x^2 - 3x + 2kx + 1 to have 2 zeros, the discriminant should be positive. The discriminant is given by b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation. In this case, the discriminant is:

(-3)^2 - 4(4)(1)(2k) = 9 - 32k

To have 2 zeros, the discriminant should be positive. So, we can set up the inequality:

9 - 32k > 0

Simplifying this inequality, we get:

-32k > -9

Dividing both sides by -32 and reversing the inequality sign (because we're dealing with a negative coefficient), we have:

k < 9/32

Therefore, the values of k for which the function f(x) = 4x^2 - 3x + 2kx + 1 have 2 zeros are k < 9/32.

And remember, never trust an atom - they make up everything!

To determine the values of k for which the function f(x) = kx^2 - 4x + k has no zeros, we can use the discriminant of the quadratic equation. The discriminant is given by b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.

In this case, the quadratic equation is kx^2 - 4x + k = 0, so a = k, b = -4, and c = k.

For the equation to have no zeros, the discriminant must be negative (less than zero), since a negative discriminant indicates that the quadratic equation has no real solutions.

Therefore, we have:

(-4)^2 - 4(k)(k) < 0
16 - 4k^2 < 0
-4k^2 + 16 < 0
k^2 - 4 > 0

To solve this inequality, we can factor it:

(k + 2)(k - 2) > 0

The quadratic will be positive when either both factors are positive or both factors are negative.

Case 1: (k + 2) > 0 and (k - 2) > 0
k > -2 and k > 2
The values of k satisfying this condition are k > 2.

Case 2: (k + 2) < 0 and (k - 2) < 0
k < -2 and k < 2
The values of k satisfying this condition are k < -2.

Therefore, the values of k for which the function f(x) = kx^2 - 4x + k has no zeros are k < -2 or k > 2.

Now, let's determine the values of k for which the function f(x) = 4x^2 - 3x + 2kx + 1 has 2 zeros.

For a quadratic equation to have two real and distinct zeros, the discriminant must be positive (greater than zero).

Given the quadratic equation, a = 4, b = -3, and c = (2k + 1).

The discriminant is given by:

b^2 - 4ac > 0
(-3)^2 - 4(4)(2k + 1) > 0
9 - 32(2k + 1) > 0
9 - 64k - 32 > 0
-64k - 23 > 0
-64k > 23
k < -23/64

Therefore, the values of k for which the function f(x) = 4x^2 - 3x + 2kx + 1 has two zeros are k < -23/64.

To determine the values of k for which the function f(x) = kx^2 - 4x + k has no zeros, we can use the discriminant of the quadratic equation.

The quadratic equation is given by ax^2 + bx + c = 0, where in our case, a = k, b = -4, and c = k.

The discriminant (D) of a quadratic equation is calculated as D = b^2 - 4ac.

For the function f(x) = kx^2 - 4x + k to have no zeros, the discriminant D must be negative since the equation does not intersect the x-axis.

Substituting the values into the discriminant formula, we get D = (-4)^2 - 4(k)(k).

Simplifying, we have D = 16 - 4k^2.

To have no zeros, D < 0, so 16 - 4k^2 < 0.

Now, let's solve for k:
16 - 4k^2 < 0
Dividing both sides by 4, we get:
4 - k^2 < 0
Rearranging, we have:
k^2 - 4 > 0
(k - 2)(k + 2) > 0

To satisfy this inequality, either both factors (k - 2) and (k + 2) must be positive, or both must be negative.

Case 1: (k - 2) > 0 and (k + 2) > 0
This implies that k > 2 and k > -2, which means k must be greater than 2.

Case 2: (k - 2) < 0 and (k + 2) < 0
This implies that k < 2 and k < -2, which means k must be less than -2.

Therefore, the function f(x) = kx^2 - 4x + k has no zeros when k > 2 or k < -2.

Now, let's move to the second part of the question.

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 again.

The quadratic equation is given by ax^2 + bx + c = 0, where in our case, a = 4, b = -3, and c = 2k + 1.

The discriminant (D) is calculated as D = b^2 - 4ac.

Substituting the values, we have D = (-3)^2 - 4(4)(2k + 1).

Simplifying, we get D = 9 - 32(2k + 1).

To have two zeros, the discriminant D must be greater than zero since the quadratic equation intersects the x-axis twice.

So, 9 - 32(2k + 1) > 0.

Simplifying further, we have 9 - 64k - 32 > 0.

Combining like terms, we get -64k - 23 > 0.

Solving for k, we have -64k > 23.

Dividing both sides by -64 (and flipping the inequality sign), we get:
k < 23/64.

Therefore, the values of k for which the function f(x) = 4x^2 - 3x + 2kx + 1 have two zeros are k < 23/64.

if b^2 - 4ac < 0 ---> no real zeros

if b^2 - 4ac = 0 ---> one real zero
ifb^2 - 4ac > 0 ---> two real zeros

if will do the 2nd question.
a=4, b= 2k-3 , c = 1

b^2 - 4ac
= (2k-3)^2 - 4(4)(1)
= 4k^2 - 12k + 9 - 16
= 4k^2 - 12k - 7
= (k + 1)(4k - 7)

(k+1)(4k-7) > 0 for 2 real roots
k < -1 OR k > 7/4