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The only solution of the equation x^2 +bc+16=0 is x=4 why is the value of b

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Do you mean bc as the middle term, or is it bx?

If it has only a solution of x = 4, then you must have

(x - 4)(x - 4) = 0

x^2 - 8x + 16 = 0

-8x = bx

b = -8
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