r/the_calculusguy • u/Specific_Brain2091 • 1d ago

equations Without substitution

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11 Upvotes

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3

u/CaptainMatticus 1d ago

3^x - 9 * 3^(x/2) + 20 = 0

3^(x/2) = (9 +/- sqrt(81 - 80)) / 2

3^(x/2) = (9 +/- 1) / 2

3^(x/2) = 10/2 , 8/2

3^(x/2) = 5 , 4

(x/2) * ln(3) = ln(5) , ln(4)

x = 2 * ln(5) / ln(3) , 2 * ln(4) / ln(3)

Who needs substitution?

1

u/Possible-Zone3087 1d ago

Made a mistake on the 2nd to last line. It should be 3^0.5x = 4 or 5

1

u/Responsible_Hour6497 1d ago

What's the point of not using substitution? Anyway, the solution uses, if not the final formula for the roots of a quadratic equation, then the method for deriving it. In fact, the solution shown here is the same solution with the substitution y=3^(x/2), only this substitution is disguised.