Given
color(white)("XXX")9x^2-12x+4=-3
There are several ways to solve this.
I will demonstrate using a "completing the square method"
The given equation implies
color(white)("XXX")9x^2-12x=-7 (after subtracting 4 from both sides)
color(white)("XXX")9(x^2-4/3x)=-7
color(white)("XXX")9(x^2-4/3x+(2/3)^2)=-7+9 * (2/3)^2
color(white)("XXX")9(x-2/3)^2=-7+4
color(white)("XXX")(x-2/3)^2=-3/9=-1/3
color(white)("XXX")(x-2/3)=+-sqrt(-1/3)
Note that for Real values sqrt(-1/3) is undefined,
but if we are allowed Complex values:
color(white)("XXX")x-2/3=+-1/sqrt(3)i
and
color(white)("XXX")x=2/3+-1/sqrt(3)i=(2+-sqrt(3)i)/3
Why is there not Real solution?
Note that the given equation is equivalent to 9x^2-12x+7=0
and here is a graph of 9x^2-12x+7
graph{9x^2-12x+7 [-12.41, 12.9, -4.42, 8.22]}
Notice that 9x^2-12x+7 never crosses the X-axis and therefore it is never equal to 0
