Question 142976
{{{x^2-3x=18}}} Start with the given equation



{{{x^2-3x+9/4=18+9/4}}} Take half of the x coefficient 3 to get {{{3/2}}}. Now square {{{3/2}}} to get {{{9/4}}}. Add this number to both sides



{{{x^2-3x+9/4=81/4}}} Add 18 and {{{9/4}}} to get {{{81/4}}}



{{{(x-3/2)^2=81/4}}} Factor the left side. Now the left side is a complete square



{{{x-3/2=0+-sqrt(81/4)}}} Take the square root of both sides



{{{x-3/2=9/2}}} or {{{x-3/2=-9/2}}} Simplify and expand



{{{x=9/2+3/2}}} or {{{x=-9/2+3/2}}} Add {{{3/2}}} to both sides in each case



{{{x=12/2}}} or {{{x=-6/2}}} Combine like terms



{{{x=6}}} or {{{x=-3}}} Reduce



So the solutions are {{{x=6}}} or {{{x=-3}}}



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a)


{{{(x-3/2)^2=81/4}}} Start with the completed square



{{{(x-3/2)^2-81/4=0}}} Subtract {{{81/4}}} from both sides


Now the equation is in vertex form {{{a(x-h)^2+k}}} where the vertex is (h,k)


So in this case, the vertex is *[Tex \LARGE \left(\frac{3}{2},-\frac{81}{4}\right)]



b)


The line of symmetry is simply the equation {{{x=h}}} where h is the x-coordinate of the vertex. So the axis of symmetry is {{{x=3/2}}} 



c)


Remember, we're still using the general form {{{a(x-h)^2+k}}}, 

Looking at {{{(x-3/2)^2-81/4}}}, we can see that {{{a=1}}}. Since {{{a>0}}}, this means that the parabola opens up and there is a minimum. So the minimum occurs at the vertex which means that the minimum is {{{-81/4}}}



d) 


Finally here's a sketch to visually verify our answers


{{{graph(500,500,-20,20,-22,22,(x-3/2)^2-81/4)}}}