Begin by setting the equation = to 0, and then move the 6 over by subtraction. Here's what it is now: [tex] \frac{3}{4} x^{2} +6x=-6 [/tex]. Now we are ready to complete the square on the x terms. First, though, the rule is that the leading coefficient has to be a +1. Ours is a 3/4. So we will factor it out. When we do that, we get [tex] \frac{3}{4} ( x^{2} +8x)=-6[/tex]. Take half the linear term, square it and add it to both sides. Our linear term is 8. Half of 8 is 4, and 4 squared is 16. So add 16 to the left inside the parethesis...BUT don't forget about that 3/4 out front there, refusing to be ignored. It is a multiplier. So what we have really added in is (3/4)(16), which is 12. Now here's what we have: [tex] \frac{3}{4}( x^{2} +8x+16)=-6+12 [/tex]. Simplifying the right side we have [tex] \frac{3}{4} ( x^{2} +8x+16)=6[/tex]. The whole point of this is to create a perfect square binomial on the left which will serve as the h in our vertex (the x-coordinate). That binomial is this: [tex] \frac{3}{4} (x+4)^2=6[/tex]. Now we will move the 6 back over by subtraction and set it back equal to y. [tex]y= \frac{3}{4} (x+4)^2-6[/tex]. The minimum value is reflected in the k value of the vertex (the y-coordinate), which is -6. So our answer is A from above.
