# Det. the quadratic fun. that has the given roots and passes through thegiven point Ex. each fun. in standard forma) x=2±√5,(2,10) b)x=1/3andx=-4,(1,5)fun.= "function" Ex.= "Express" Det.=...

Det. the quadratic fun. that has the given roots and passes through the
given point Ex. each fun. in standard forma) x=2±√5,(2,10)

b)x=1/3andx=-4,(1,5)

fun.= "function"

Ex.= "Express"

Det.= "Determine"

embizze | High School Teacher | (Level 1) Educator Emeritus

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(a) Find the equation for the quadratic that passes through (2,10) and has `2+-sqrt(5)` as roots.

(1) If `2+sqrt(5)` is a root, then `(x-(2+sqrt(5)))` is a factor. Similarly, if `2-sqrt(5)` is a root, then `(x-(2-sqrt(5)))` is a factor. Thus we have:

`f(x)=a(x-2-sqrt(5))(x-2+sqrt(5))=a(x^2-4x-1)`

(2)When x=2, f(x)=10 so:

`a((2)^2-4(2)-1)=10=>a=-2`

Thus `f(x)=-2(x^2-4x-1)=-2x^2+8x+2`

(b) Find the equation of the quadratic function that passes through (1,5) and has roots of 1/3 and -4.

(1) If 1/3 is a root, (x-1/3) is a factor. Also if x=-4 is a root, then (x+4) is a factor. So `f(x)=a(x-1/3)(x+4)`

(2) When x=1, f(x) =5 so:

`a(1-1/3)(1+4)=5=>a=3/2`

Then `f(x)=3/2(x-1/3)(x+4)`

`=3/2[x^2+(11)/3x-4/3]`

`=3/2 x^2+(11)/2x-2`