If f(x)=x^2-6x+14 and g(x)=-x^2-20x-k, determine the value of k so that there is exactly one point of intersection between the two parabolas.
then
x^2 - 6x + 14 = -x^2 - 20x - k
2x^2 + 14x + 14+k = 0
in this quadratic
a = 2
b= 14
c = 14+k
to have only one solution (root), the value of the discriminant has to be zero, thus
b^2 - 4ac = 0
14^2 - 4(2)(14+k) = 0
solve for k
Thanks!
Why did the parabola go see a therapist? Because it had too many intersecting issues! Now, let's find that special value of k.
To find the point of intersection, we need to set f(x) equal to g(x). So, we have:
x^2 - 6x + 14 = -x^2 - 20x - k
Now, let's simplify this equation by moving everything to one side:
2x^2 - 14x + 14 + 20x + k = 0
Combine like terms:
2x^2 + 6x + 14 + k = 0
For there to be exactly one point of intersection, the quadratic equation should have a discriminant (b^2 - 4ac) of zero. So, let's set up the equation:
b^2 - 4ac = 0
(6)^2 - 4(2)(14 + k) = 0
36 - 112 - 8k = 0
-8k - 76 = 0
Now, solve for k:
-8k = 76
k = -76/8
Divide and conquer:
k = -9.5
Therefore, the value of k that gives exactly one point of intersection is -9.5.
To find the value of k so that there is exactly one point of intersection between the two parabolas, we need to set the two functions equal to each other and solve for x.
Setting f(x) equal to g(x), we have:
x^2 - 6x + 14 = -x^2 - 20x - k
To simplify the equation, let's gather all the terms on one side:
x^2 + x^2 - 6x + 20x + 14 + k = 0
Combining like terms, we get:
2x^2 + 14x + 14 + k = 0
Now, we need to find the discriminant of this quadratic equation to determine the number of solutions. The discriminant (D) can be calculated using the formula D = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation (ax^2 + bx + c = 0).
In this case, a = 2, b = 14, and c = (14 + k). Substituting these values into the formula:
D = (14)^2 - 4(2)(14 + k)
= 196 - 8(14 + k)
= 196 - 112 - 8k
= 84 - 8k
For there to be exactly one point of intersection, the discriminant D must be equal to zero. So, we have:
84 - 8k = 0
Solving for k:
8k = 84
k = 84/8
k = 10.5
Therefore, the value of k that results in exactly one point of intersection between the two parabolas is 10.5.
To find the value of k that results in exactly one point of intersection between the two parabolas, you need to set the two functions equal to each other and solve for x.
The equation will be: f(x) = g(x)
Substituting the given functions, we have: x^2 - 6x + 14 = -x^2 - 20x - k
Next, rearrange this equation to get all the terms on one side:
x^2 + x^2 - 6x + 20x + 14 + k = 0
Combine like terms:
2x^2 + 14x + 14 + k = 0
For there to be exactly one point of intersection, the quadratic equation 2x^2 + 14x + 14 + k = 0 must have a single solution. This means that the discriminant, b^2 - 4ac, should be equal to zero.
Using the equation: discriminant = b^2 - 4ac
Substituting the coefficients from the quadratic equation:
(14)^2 - 4(2)(14 + k) = 0
Expanding and simplifying this equation:
196 - 56(14 + k) = 0
196 - 784 - 56k = 0
-588 - 56k = 0
Now, solve for k:
-56k = 588
k = -588/-56
k = 10.5
Therefore, the value of k that results in exactly one point of intersection between the two parabolas is k = 10.5.