Question 1207659
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Find the real solutions of the equation.

4x^(1/2) - 9x^(1/4) + 4 = 0
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<pre>
Introduce new variable  y = x^(1/4).

Then our original equation takes the form

    4y^2 - 9y + 4 = 0.


It is a quadratic equation for y.  Solve it using the quadratic formula

    {{{y[1,2]}}} = {{{(9 +- sqrt((-9)^2 - 4*4*4))/(2*4)}}} = {{{(9 +- sqrt(81-64))/8}}} = {{{(9 +- sqrt(17))/8}}}.


Thus, there are two real roots for y:  y= {{{(9 + sqrt(17))/8}}}  and  y= {{{(9 - sqrt(17))/8}}}.


It gives two real positive solutions for x

    x = y^4 = {{{((9 + sqrt(17))/8)^4}}} = 7.240799963  (approximately).

and

    x = y^4 = {{{((9 - sqrt(17))/8)^4}}} = 0.138106287  (approximately).
</pre>

Solved.


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Again, the method of solution is changing the variable.