determine the value of k such that g(x)=3x+k intersects the quadratic function f(x)+2x^2-5x+3
we want 2x^2 - 5x + 3 = 3x + k
2x^2 - 8x + 3-k = 0
Now, this equation has solutions as long as the discriminant is not negative
64 - 8(3-k) >= 0
40 + 3k >= 0
So, you can pick any value for k >= -40/3.
Think of the graphs. You have a parabola that open up, and a line with slope=3. If the line is too low, it will never touch the parabola. Raise it high enough, and it will always intersect the parabola, which extends upward forever.
Now, if you want a specific k, then you need to specify where the line intersects, or the slope at the intersection, or something.
To find the value of k that makes the linear function g(x) intersect the quadratic function f(x), we need to find the x-value at which they intersect.
First, let's set the two functions equal to each other:
f(x) = g(x)
2x^2 - 5x + 3 = 3x + k
Next, rearrange the equation to put it in standard quadratic form:
2x^2 - 8x + (3 - k) = 0
Now, we have a quadratic equation. To find the x-values at which the two functions intersect, we can either factorize the quadratic equation or use the quadratic formula.
Option 1: Factorize the quadratic equation:
To factorize the equation, we need to find two numbers whose sum is -8 and product is (2)(3 - k). Factoring the quadratic equation in this case might not be straightforward, as we don't know the value of k yet.
Option 2: Use the quadratic formula:
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In the case of our equation, the values of a, b, and c are:
a = 2
b = -8
c = 3 - k
Plugging these values into the quadratic formula, we get:
x = (-(-8) ± √((-8)^2 - 4(2)(3 - k))) / (2(2))
x = (8 ± √(64 - 24 + 8k)) / 4
x = (8 ± √(40 + 8k)) / 4
x = (8 ± √(8(5 + k))) / 4
x = (8 ± 2√(5 + k)) / 4
Simplifying further:
x = 2 ± 0.5√(5 + k)
Now, to find the x-values where the two functions intersect, we solve for x. This means that the discriminant (the expression inside the square root) must be greater than or equal to zero:
5 + k ≥ 0
Solving for k:
k ≥ -5
Therefore, the value of k that makes the two functions intersect is any value greater than or equal to -5.