Find a so that ax^2 + 15x + 4 + 3x + 5 is the square of a binomial.

 Oct 17, 2021


Find a so that ax^2 + 15x + 4 + 3x + 5 is the square of a binomial.  


First, combine like terms.  This means add 15x and 3x  and  add 4 and 5  


ax2 + 18x + 9   


So let's factor this quadratic equation.  

I recognize that this particular one will be simple.  They won't all be simple.  

Remember that, since we're looking for a square, both factors will be the same.  


Write down the parentheses first.                                          (            )(            )   


Since 18x is positive, there will be a + between the terms      (     +     )(     +     )   


The second term inside the parentheses will be the 

square root of the final term in the orginal equation               (     +   3)(     +   3)    


Now the easy part is done.  The first term inside the   

parentheses will be the square root of the first term   

in the original equation.  Our job is to figure out the   

square root of "a" that will produce 18 in the middle.  


Each factor will have two terms.  We want a number    

to go on the left, so that when the product of the   

outside terms is added to the product of the inside  

terms, the sum is 18.  Forget the "x" for now, that  

will take care of itself.  I don't know how to tell you    

an easy way to do this.  I just look at it and try a   

couple of likely values.  Cut to the chase, it's 3.                      (3x  +   3)(3x  +   3)


So the quadratic is 9x2 + 18x + 9, and the answer asked for in the problem is a = 9.    


 Oct 17, 2021

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