Help me with this math question!!
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Help me with this math question!!

[From: ] [author: ] [Date: 11-08-22] [Hit: ]
......
The Parabola P has equation y =- x ^ 2 +2 x-3 and the function f has the rule f (x) = 2x ^ 2-8x + k

a) Determine the intersection between P and f

b) Determine k so that the graphs of f and P has exactly one common point

Please help me and explain! Thanks a lot..

-
y = -x² + 2x - 3
f(x) = 2x² - 8x + k

P and f intersect when
2x² - 8x + k = -x² + 2x - 3
3x² - 10x + k+3 = 0

Use the quadratic equation to solve for x
x = [10±√(10² - 4·3(k+3))]/(2·3)
= [10±√(100 - 12k-36)]/6
= [10±√(64 - 12k)]/6

There is one value for x when 64 - 12k = 0
k = 5⅓

P and f intersect at exactly one point when k = 5⅓.
The x-coordinate of the point of intersection is
x = [10±√(64 - 12(5⅓))]/6
= 10/6
= 1⅔
The y-coordinate of the point of intersection is
y = -(1⅔)² + 2(1⅔) - 3 = -22/9

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