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Find all real values of $s$ such that $x^2 + sx + 144 - 63 + x^2 - 25 - 18$ is the square of a binomial.

 May 19, 2024
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x^2 + sx + 144 - 63 + x^2 - 25 - 18

 

combine like terms: 

2x^2 + sx + 38

 

if this is a square of a binomial then the first term must be (sqrt2 * x)

2x^2 + sx + 38 = (sqrt2 * x + [other term])^2

 

this other term has to be sqrt 38 because the contant term of the expression is 38

2x^2 + sx + 38 = (sqrt2 * x + sqrt38)^2

 

sx is 2ab according to binomial square expansion: (a+b)^2 = a^2 + 2ab + b^2

a = sqrt2 * x

b = sqrt38

so 2ab = sqrt76 * x

so s = sqrt76 = 2sqrt19

 May 19, 2024

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