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Roots-of-the-equation-9x-2-18-x-5-0-belonging-to-the-domain-of-definition-of-the-function-f-x-log-x-2-x-2-is-are-

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Question Number 4882 by Rishabh Jain last updated on 19/Mar/16
Roots of the equation 9x^2 −18∣x∣+5=0 belonging to the domain of definition of the function f(x)=log (x^2 −x−2) is/ are
Rootsoftheequation9x2−18∣x∣+5=0belongingtothedomainofdefinitionofthefunctionf(x)=log(x2−x−2)is/are
Answered by Yozzii last updated on 19/Mar/16
We are given that 9x^2 −18∣x∣+5=0 (∗) Since x^2 =∣x∣^2 , (∗) becomes 9∣x∣^2 −18∣x∣+5=0 with discriminant D valued as D=18^2 −4×9×5>0. So, the roots of (∗) are real and distinct. Factorising (∗) gives (3∣x∣−1)(3∣x∣−5)=0. Hence, x=((±1)/3),((±5)/3). Now, the logarithmic function f(x)=log(h(x)) is real valued iff the range of h(x) is strictly positive. So, let h(x)=x^2 −x−2=(x+1)(x−2). h(x)>0 when (1) x+1<0 & x−2<0 ⇒ x<−1, or (2) when x+1>0 & x−2>0 ⇒x>2. The domain of f(x) is the set A={x∈R∣x>2 or x<−1}. The only solution that lies in A is ((−5)/3). So, the required value of x is x=((−5)/3).
Wearegiventhat9x2−18∣x∣+5=0(∗)Sincex2=∣x∣2,(∗)becomes9∣x∣2−18∣x∣+5=0withdiscriminantDvaluedasD=182−4×9×5>0.So,therootsof(∗)arerealanddistinct.Factorising(∗)gives(3∣x∣−1)(3∣x∣−5)=0.Hence,x=±13,±53.Now,thelogarithmicfunctionf(x)=log(h(x))isrealvaluedifftherangeofh(x)isstrictlypositive.So,leth(x)=x2−x−2=(x+1)(x−2).h(x)>0when(1)x+1<0&x−2<0⇒x<−1,or(2)whenx+1>0&x−2>0⇒x>2.Thedomainoff(x)isthesetA={x∈R∣x>2orx<−1}.TheonlysolutionthatliesinAis−53.So,therequiredvalueofxisx=−53.

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