Determin the points of intersection algebraically :
f(x)=-x(squared) +6x-5, g(x)=-4x+19
To determine the points of intersection between two functions algebraically, you need to find the x-values at which the two functions are equal. Then, you can substitute these x-values back into either one of the functions to find the corresponding y-values.
In this case, we have two functions f(x) and g(x). We need to find the points where f(x) and g(x) intersect, which means we need to find the x-values at which f(x) = g(x).
Step 1: Set the two functions equal to each other.
-f(x) + 6x - 5 = -4x + 19
Step 2: Simplify the equation.
-x^2 + 6x - 5 = -4x + 19
Step 3: Rearrange the equation into standard quadratic form (ax^2 + bx + c = 0).
-x^2 + 6x + 4x - 5 - 19 = 0
Step 4: Combine like terms.
-x^2 + 10x - 24 = 0
Step 5: Solve the equation by factoring, completing the square, or using the quadratic formula.
Since the given equation can be factored, let's try factoring it.
Step 6: Factor the equation.
-(x - 4)(x - 6) = 0
Setting each factor equal to zero:
x - 4 = 0 => x = 4
x - 6 = 0 => x = 6
Therefore, the x-values at which f(x) = g(x) are x = 4 and x = 6.
Step 7: Substitute each x-value back into either f(x) or g(x) to find the corresponding y-values.
Let's substitute these x-values into f(x).
When x = 4:
f(4) = -(4)^2 + 6(4) - 5
= -16 + 24 - 5
= 3
When x = 6:
f(6) = -(6)^2 + 6(6) - 5
= -36 + 36 - 5
= -5
Therefore, the points of intersection between f(x) and g(x) are (4, 3) and (6, -5).
you are solving
-x^2 + 6x - 5 = -4x + 19
-x^2 + 10x - 24 = 0
x^2 - 10x + 24 = 0
(x-6)(x-4) = 0
x = 6 or x = 4
find g(6) and g(4) for the corresponding y values of the two intersection points