Get 2 FREE Instant-Explanations on Filo with code FILOAPP

The equation {x}^{{{({{3}}{{4}})}{({{}_{{2}}{x}})}^{{2}}+{({{}

Question

The equation $x^{(\frac{3}{4}) (l o g_{2} x)^{2} + (l o g_{2} x) - (\frac{5}{4})} = 2$ has $(1)$ at least one real solution $(2)$ exactly three solutions $(3)$ exactly one irrational solution $(4)$ complex roots

Views: 1,036 students

Found 6 tutors discussing this question

Charlotte Discussed

The equation

x^{(\frac{3}{4}) (l o g_{2} x)^{2} + (l o g_{2} x) - (\frac{5}{4})} = 2

has

(1)

at least one real solution

(2)

exactly three solutions

(3)

exactly one irrational solution

(4)

complex roots

5 mins ago

Discuss this question LIVE

5 mins ago

Text solutionVerified

Solution:

x^{(\frac{3}{4}) (l o g_{2} x)^{2} + l o g_{2} x - \frac{5}{4}} = 2

Taking logs both sides,

\Rightarrow ((\frac{3}{4}) (lo g_{2} x)^{2} + lo g_{2} x - \frac{5}{4}) lo g_{2} x = lo g_{2} 2^{\frac{1}{2}}

\Rightarrow ((\frac{3}{4}) (lo g_{2} x)^{2} + lo g_{2} x - \frac{5}{4}) lo g_{2} x = \frac{1}{2}

Let

lo g_{2} x = t

Then, equation becomes,

\Rightarrow (\frac{3}{4} t^{2} + t - \frac{5}{4}) t = \frac{1}{2}

\Rightarrow 3 t^{3} + 4 t^{2} - 5 t - 2 = 0

\Rightarrow (t - 1) (3 t^{2} + 7 t + 2) = 0

\Rightarrow (t - 1) (3 t^{2} + 6 t + t + 2) = 0

\Rightarrow (t - 1) (t + 2) (3 t + 1) = 0

\Rightarrow t = 1 or t = - 2 or t = - \frac{1}{3}

\Rightarrow lo g_{2} x = 1 or lo g_{2} x = - 2 or lo g_{2} x = - \frac{1}{3}

\Rightarrow x = 2^{1}, 2^{- \frac{1}{3}}, 2^{- 2}

So, there are exactly three solutions for

x

Was this solution helpful?

150

Report

One destination to cover all your homework and assignment needs

Learn Practice Revision Succeed

Instant 1:1 help, 24x7

60, 000+ Expert tutors

Textbook solutions

Big idea maths, McGraw-Hill Education etc

Essay review

Get expert feedback on your essay

Schedule classes

High dosage tutoring from Dedicated 3 experts

Trusted by 4 million+ students

Stuck on the question or explanation?

Connect with our math tutors online and get step by step solution of this question.

231 students are taking LIVE classes

Question Text	The equation $x^{(\frac{3}{4}) (l o g_{2} x)^{2} + (l o g_{2} x) - (\frac{5}{4})} = 2$ has $(1)$ at least one real solution $(2)$ exactly three solutions $(3)$ exactly one irrational solution $(4)$ complex roots
Answer Type	Text solution:1
Upvotes	150