Question

How do you find the vertex and the intercepts for `f(x)= -6x^2+ 5x + 18`?

Answer 1

`"vertex "=(5/12,457/24),x=5/12+-sqrt457/12`

Explanation:

`"the equation of a parabola in "color(blue)"vertex form"` is.

`•color(white)(x)y=a(x-h)^2+k`

`"where "(h,k)" are the coordinates of the vertex and a"`
`"is a multiplier"`

`"to obtain this form "color(blue)"complete the square"`

`y=-6(x^2-5/6x-3)`

`color(white)(y)=-6(x^2+2(-5/12)x+25/144-25/144-3)`

`color(white)(y)=-6(x-5/12)+457/24larrcolor(blue)"in vertex form"`

`color(magenta)"vertex "=(5/12,457/24)`

`"to obtain the x-intercepts set y = 0"`

`-6(x-5/12)^2+457/24=0`

`(x-5/12)^2=457/144`

`color(blue)"take the square root of both sides"`

`x-5/12=+-sqrt(457/144)=+-sqrt457/12`

`"add "5/12" to both sides"`

`x=5/12+-sqrt457/12larrcolor(red)"exact values"`