The parabolas y = -3x2 + 10x - 6 and y = -3x2 - 17x + 2 intersect at _____?
A.the line y = 27x - 8.
B. x=the square root of 2 over 3 and xthe -square root of 2 over 3.
C. x=8/27.
D. they do not intersect.
two curves intersect in a point, so
A is out
solving the equation
-3x^2 + 10x - 6 = -3x^2 - 17x + 2
yields x = 8/27
- 3 x ^ 2 + 10 x - 6 = - 3 x ^ 2 - 17 x + 2
Add 3 x ^ 2 + 17 x - 2 to both sides.
- 3 x ^ 2 + 3 x ^ 2 + 10 x + 17 x - 6 - 2 = - 3 x ^ 2 + 3 x ^ 2 - 17 x + 17 x + 2 - 2
27 x - 8 = 0
Add 8 to both sides.
27 x - 8 + 8 = 0 + 8
27 x = 8 Divide both sides by 27
x = 8 / 27
when x = 8 / 27
- 3 x ^ 2 + 10 x - 6 = - 3 * ( 8 / 27 ) ^ 2 + 10 * 8 / 27 -6 =
- 3 * 64 / 729 + 10 * 8 / 27 - 6 =
- 192 / 729 + 80 / 27 - 6 =
- 192 / 729 + 27 * 80 / 729 + 6 * 729 / 729 =
- 192 / 729 + 2160 / 729 - 4374 / 729 =
- 2406 / 729 = - 3 * 802 / ( 3 * 243 ) =
- 802 / 243
Intersection point ( 8 / 27 , - 802 / 243 )
Thank youu! (:
To find the points of intersection between the two parabolas, we need to set their respective equations equal to each other.
So we have:
-3x^2 + 10x - 6 = -3x^2 - 17x + 2
Now, let's simplify this equation:
We can start by eliminating the -3x^2 terms on both sides:
10x - 6 = -17x + 2
Next, let's combine like terms:
10x + 17x = 2 + 6
27x = 8
Now, divide both sides by 27 to solve for x:
x = 8/27
So the value of x where the two parabolas intersect is x = 8/27.
To find the corresponding y-value, substitute this value of x back into either of the original equations. Let's use the first equation:
y = -3(8/27)^2 + 10(8/27) - 6
After simplifying, we get:
y = -192/729 + 80/27 - 6
Finding a common denominator and simplifying:
y = -192/729 + 2160/729 - 4374/729
y = -3996/729
Simplifying further, we have:
y = -36/729
Now, we can conclude that the two parabolas intersect at the point (8/27, -36/729).
Going back to the answer choices, we can see that none of them matches the point of intersection we found. Therefore, the correct answer is:
D. They do not intersect.