# What would be the best method to solve #-3x^2+12x+1=0#?

##### 1 Answer

#### Answer:

#### Explanation:

Given:

#-3x^2+12x+1 = 0#

This is in the form:

#ax^2+bx+c = 0#

with

In order to decide which method to use, we can first examine the discriminant:

#Delta = b^2-4ac = color(blue)(12)^2-4(color(blue)(-3))(color(blue)(1)) = 144+12 = 156#

Since

So there is no rational factorisation and we can discount the use of an AC method or similar.

We can use the quadratic formula or we can complete the square.

If we choose to use the quadratic formula, then we find:

#x = (-b+-sqrt(b^2-4ac))/(2a)#

#color(white)(x) = (-b+-sqrt(Delta))/(2a)#

#color(white)(x) = (-(color(blue)(12))+-sqrt(156))/(2(color(blue)(-3)))#

#color(white)(x) = (-12+-2sqrt(39))/(-6)#

#color(white)(x) = 2+-sqrt(39)/3#

When completing the square, note that

#0 = -3(-3x^2+12x+1)#

#color(white)(0) = 9x^2-36x-3#

#color(white)(0) = (3x)^2-2(3x)(6)+(6)^2-39#

#color(white)(0) = (3x-6)^2-(sqrt(39))^2#

#color(white)(0) = ((3x-6)-sqrt(39))((3x-6)+sqrt(39))#

#color(white)(0) = (3x-6-sqrt(39))(3x-6+sqrt(39))#

So:

#3x = 6+-sqrt(39)#

So:

#x = 2+-sqrt(39)/3#