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Find the sum of all values of $k$ for which $x^2 + kx - 9x + 25 - 9$ is the square of a binomial.

 Jul 4, 2024
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Find the sum of all values of $k$ for which $x^2 + kx - 9x + 25 - 9$ is the square of a binomial.   

 

                                                                       x2 + (kx – 9x) + 16   

 

factor out x                                                      x2 + (k–9)(x) + 16

 

When posed as ax2 + bx + c, for there to be     

a double root, also called a repeated root,   

the term c must be a square and b = 2sqrt(c)  

 

Square root of 16 = +4 so b = (2)(+4) = +8                  

 

when +4 is positive                                         k – 9 = +8        

                                                                       k = +8 + 9   

                                                                       k = 17  

 

when +4 is negative                                        k – 9 = –8   

                                                                       k = –8 + 9   

                                                                       k = 1   

 

the sum of all values of k                                sum = 17 + 1   

                                                                       sum = 18   

.

 Jul 4, 2024

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