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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.

 Jun 11, 2024
 #1
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Combining all like terms in the equation, 

\(x^2 + (k-9)x + 16\)

 

In order for this to be the square of a binomial, the coefficient of x has to be 2 times the square root of the third term. 

 

Take this

\((x+b)^2=x^2+2bx+b^2\)

 

2b is exactly two times the square root of the third term. 

 

In this case, we have \(k-9=2\sqrt{16}\)

 

We have

\(k-9=8\\ k=17\)

 

So k is 17. 

 

Thanks! :)

 Jun 11, 2024

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