the graph of the function
f(x)=x^2-kx+k+8 touches the x-axis at one point. What are the possible values of k
see previous answer. You need
x^2 - kx + k+8 to be a perfect square
To find the possible values of k for which the graph of the function touches the x-axis at one point, we need to determine the discriminant of the quadratic equation formed by setting the function equal to zero.
The given function is:
f(x) = x^2 - kx + k + 8
Setting it equal to zero gives:
x^2 - kx + k + 8 = 0
The discriminant (D) of a quadratic equation ax^2 + bx + c = 0 is given by:
D = b^2 - 4ac
In this case, a = 1, b = -k, and c = k + 8.
Therefore, the discriminant is:
D = (-k)^2 - 4(1)(k + 8)
= k^2 - 4(k + 8)
For the graph to touch the x-axis at one point, the discriminant should be equal to zero. So, let's set D = 0 and solve for k:
k^2 - 4(k + 8) = 0
Expanding and rearranging the equation, we get:
k^2 - 4k - 32 = 0
Now we can solve this equation either by factoring, completing the square, or using the quadratic formula. Let's solve it by factoring:
(k - 8)(k + 4) = 0
Setting each factor equal to zero gives:
k - 8 = 0 or k + 4 = 0
Solving for k in each equation gives us two possible values:
k = 8 or k = -4
Therefore, the possible values of k for which the graph of the function touches the x-axis at one point are k = 8 and k = -4.
To find the possible values of k for which the graph of the function f(x) = x^2 - kx + k + 8 touches the x-axis at one point, we need to consider the discriminant of the quadratic equation formed by equating f(x) to zero.
The discriminant, denoted as D, is a part of the quadratic formula and is calculated as D = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
In this case, the coefficients of the quadratic equation are a = 1, b = -k, and c = k + 8. Therefore, the discriminant can be written as D = (-k)^2 - 4(1)(k + 8).
For the graph of the quadratic to touch the x-axis at one point, the discriminant D must be equal to zero. Mathematically, this can be represented as D = 0.
So, in order to find the possible values of k, we can solve the equation (-k)^2 - 4(1)(k + 8) = 0 for k.
Let's solve this equation:
k^2 - 4(k + 8) = 0
Expanding and simplifying:
k^2 - 4k - 32 = 0
Now, we can use factoring, the quadratic formula, or completing the square to solve this equation for k. Let's solve it by factoring:
(k - 8)(k + 4) = 0
Setting each factor equal to zero:
k - 8 = 0 or k + 4 = 0
Solving for k in both equations:
k = 8 or k = -4
Therefore, the possible values of k for which the graph of the function touches the x-axis at one point are k = 8 and k = -4.