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Solve by completing the square; round to the nearest hundredth, if necessary

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3 Answers

Completing the Square.  Standard form of a quadratic is ax2+bx+c=0.
The first step in CtS is to subtract the "c"... but this has already been done for you.
3x2+15x=108
 
Next, divide all of the terms by "a"... here a=3.
3x2/3+15x/3=108/3
x2+5x=36
 
Next, we actually "complete the square."  We take the expression on the left and turn it into a perfect square trinomial by finding a new constant (number with no variable).  To keep the equation balanced, we will add this value to both sides of the equation.
Now, how to find it?  Take your new "b" value (in this case: +5... the sign will be important when we factor...), divide it by 2, then square it.  Word to the wise: keep it "messy" on the left.
x2+5x+(+5/2)2=36+6.25                Note: it's your choice to use 6.25 or leave it as +(5/2)2 on the right.
 
Now, factor it.  If you kept it messy on the left, take take a variable and what's in the parentheses to create your binomial.  You can check this with actual factoring techniques, and you'll see that it works.
(x+ 5/2)2=42.25
 
Use the Square Root Property.  Don't forget the ±roots when you take the square root of both sides of the equation.
x+ 5/2 = ±√42.25
x+ 5/2 = ±6.5
 
Solve for your variable.  Remember, you basically have two equations here (because of the ±).  If you need to see them, they are:
x+ 5/2 = 6.5                                   x+ 5/2 = -6.5
   - 5/2    -5/2                                     -5/2     -5/2
x          =  4                                     x         = -9
 
Regardless of which method you use to solve this particular equation, we should all get x=4 and x=-9.  These are also called the x-intercepts (if you were to graph the equation, you'd see it intersect the x-axis at these x-values).
3x^2+15x-108=0
Applying the foil method
 
3x^+27x-12x-108=0
3x(x+9)-12(x+9)
3(x+9)(x-4)=0
 
So,  x=-9   &   x=4
 
 

Comments

The instructions were to solve by completing the square.  I still think it's good to show learners that other methods will work, too, so they can use more familiar methods to check their work. :)