Question 274516
equation is:


0=3x-9√x-14?


let k = sqrt(x)


equation becomes:


0 = 3k^2 - 9k - 14


this is a quadratic equation.


factor using the quadratic formula of k = {{{(-b +- sqrt(b^2-4ac))/(2a)}}}


a = 3
b = -9
c = -14
2a = 6
-b = 9
b^2 = 81
4ac = -168
b^2 - 4ac = 81 + 168 = 249


quadratic formula becomes k = {{{(9 +- sqrt(249))/(6)}}} which becomes:


k = (9 +/- 5.77973384)/6 which becomes:


k = 4.12995564
k = -1.12995564


confirm these are correct by plugging into the equation of 3k^2 - 9k - 14 = 0.


once confirmed (done already), then you need to translate the k to x using the formula you used to derive k.


that formula was {{{k = sqrt(x)}}}


square both sides of this equation to get {{{x = k^2}}}


your answers of:


k = 4.12995564
k = -1.12995564


become:


x = 17.05653359
x = 1.276799748


plug these into the original equation of {{{3x -9*sqrt(x) - 14 = 0}}} to confirm they are good.


the value of 17.05653359 is confirmed as good since {{{sqrt(17.05653359)}}} = +/- 4.12995564 except that you have to use the positive square root only.


the value of 1.276799748 is confirmed as good since {{{sqrt(1.276799748)}}} = +/- 1.12995564 except that you have to use the negative square root only.