what the value of k
KXsq-10x-5<or=0 for all x?
k x^2 - 10 x - 5 = y
k x^2 - 10 x = y + 5
x^2 -(10/k)x = (1/k)(y+5)
x^2 -(10/k) x + 25/k^2 = y/k+ 5/k +(5/k)^2
(x-5/k)^2 = (1/k)(y+5 +25/k)
vertex at x = 5/k and y= -5-25/k
I want k negative so that the function is negative for large +or - x
I want the vertex at y = 0
-5 - 25/k = 0
25 / k = -5
k = -5
check my arithmetic !
-5 x^2 - 10 x - 5 = y
x^2 + 2 x + 1 = -y/5
(x+1)^2 = -(1/5)(y-0)
sure enough it works :)
To find the value of k for the inequality KX^2 - 10x - 5 ≤ 0 for all x, we need to solve the quadratic equation and determine the range of k that satisfies the inequality.
First, let's solve the quadratic equation:
KX^2 - 10x - 5 = 0
To solve the equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Comparing this formula to our equation (KX^2 - 10x - 5 = 0), we obtain:
a = K, b = -10, c = -5
Now, calculating the discriminant:
D = b^2 - 4ac
D = (-10)^2 - 4(K)(-5)
D = 100 + 20K
To ensure that the inequality holds true for all x, we need the discriminant to be non-positive:
D ≤ 0
100 + 20K ≤ 0
Solving this inequality, we find:
20K ≤ -100
K ≤ -5
Therefore, the value of k must be less than or equal to -5 for the inequality KX^2 - 10x - 5 ≤ 0 to hold true for all x.
To find the value of k in the given inequality: KX^2 - 10x - 5 ≤ 0 for all x, we need to determine the range of x values for which the inequality holds true.
To solve quadratic inequalities like this, we can first find the roots of the quadratic equation, which are the x-intercepts where the quadratic expression equals zero.
In this case, the quadratic equation is KX^2 - 10x - 5 = 0.
The discriminant (b^2 - 4ac) of this equation is 100 + 20K. By examining the discriminant, we can determine the nature of the roots:
1. If the discriminant is positive (greater than zero), the quadratic has two distinct real roots.
2. If the discriminant is zero, the quadratic has a repeated root.
3. If the discriminant is negative (less than zero), the quadratic has no real roots.
Let's consider each case:
Case 1: If the discriminant is positive (100 + 20K > 0)
For a quadratic inequality, if the discriminant is greater than zero, it means the quadratic expression is positive on either side of the roots. In this case, the expression KX^2 - 10x - 5 is positive for all x outside the range of the roots.
Case 2: If the discriminant is zero (100 + 20K = 0)
If the discriminant is zero, the quadratic has a repeated root. In this case, the inequality would hold true only for the values of x that make the quadratic expression zero.
Case 3: If the discriminant is negative (100 + 20K < 0)
If the discriminant is negative, the quadratic has no real roots. In this case, the quadratic expression remains either positive or negative for all x.
So, to find the value of k that satisfies the inequality KX^2 - 10x - 5 ≤ 0 for all x, we need to consider Case 1: 100 + 20K > 0.
Solving this inequality:
100 + 20K > 0
20K > -100
K > -100/20
K > -5
Thus, the value of k must be greater than -5 for the inequality KX^2 - 10x - 5 ≤ 0 to hold true for all x.