Intersection point of 2 functions?

Question

I have two functions (well, one and its inverse) and I have to find the intersection point:

y = 0.25 ( x - 4 ) ^ 2 +2
y = 2* sqrt (x-2) + 4

The answer is (6 + 2*sqrt(3), 6 + 2*sqrt(3))

Can someone show me how to get this answer? (I tried it, but it ends up being really complicated when it's not supposed to)

I tried replacing it but I just get this long thing that I don't know how to solve; i don't know if i'm doing it right

Answer 1

Other than solving a fourth degree polynomial you have to use calculus to solve.

From calculus we know that the at any given x the slope of a function and its inverse are reciprocal. Therefore if we take the derivative of one and reciprocate the derivative and set them equal we will find the x where they intersect.

y1= .25(x-4)^2+2
y2=2sqrt(x-2)+4

y1'=.25*2(x-4)
y2'=1/sqrt(x-2)

The statement above in formula form will work out like this:

y1'=1/(y2')

(1/2)(x-4)=sqrt(x-2)

(.5(x-4))^2=x-2

.25x^2-2x+4-x+2=0

.25x^2-3x+6=0

enter quadratic formula:

x= (3+Sqrt(9-(4*.25*6)))/(2*.25)

(3+sqrt(3))/(1/2)

6+2sqrt(3)

I suppose it isnt obvious when you look at it but calculus is quite handy in places like this.

Answer 2

m and n intersect the place they're equivalent –2(x + 3)(4x – 3) = –(x + 3)²(2x – 5) 2x³ – x² – 30x – 27 = 0 From the unique equation, x = –3 is a root (2x³ – x² – 30x – 27)/(x + 3) = 2x² – 7x – 9 = (2x – 9)(x + a million) = 0; x = –a million, x = 9/2 The factors of intersection are (–a million, 28), (–3, 0), and (9/2, –225)

Answer 3

its easy you just do this.

2*sqrt(x-2)+4 = .25 (x - 4)^ 2 + 2

and solve X and thats were they intersect, if they dont intersect (they would have to be parallel lines) thats when you get weird stuff. like imaginary numbers or infinities. I think... lol haha

Answer 4

Offhand, it looks like you're going to need to solve a fourth-degree polynomial.

Answer 5

Just replace and solve.