Number of solutions of equation log _{(x+1)}left(2 x^{2}+7 x+5right)+log

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4-2.Quadratic Equations and Inequations

hard

The number of solutions of the equation $\log _{(x+1)}\left(2 x^{2}+7 x+5\right)+\log _{(2 x+5)}(x+1)^{2}-4=0, x\,>\,0$, is $....$

$2$

$4$

$6$

$1$

(JEE MAIN-2021)

Solution

$\log _{(x+1)}\left(2 x^{2}+7 x+5\right)+\log _{(2 x+5)}(x+1)^{2}-4=0$

$\log _{(x+1)}(2 x+5)(x+1)+2 \log _{(2 x+5)}(x+1)=4$

$\log _{(x+1)}(2 x+5)+1+2 \log _{(2 x+5)}(x+1)=4$

$\text { Put } \log _{(x+1)}(2 x+5)=t$

$t+\frac{2}{t}=3 \Rightarrow t^{2}-3 t+2=0$

$t=1,2$

$\log _{(x+1)}(2 x+5)=1 \quad \& \log _{(x+1)}(2 x+5)=2$

$x+1=2 x+3 \quad \& \quad 2 x+5=(x+1)^{2}$

$x=-4$ (rejected) $\quad x^{2}=4 \Rightarrow x=2,-2$ (rejected)

So, $x=2$

No. of solution $=1$

Standard 11

Mathematics

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normal

(KVPY-2009)

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hard

(JEE MAIN-2024)

Let $p$ and $q$ be two real numbers such that $p+q=$ 3 and $p^{4}+q^{4}=369$. Then $\left(\frac{1}{p}+\frac{1}{q}\right)^{-2}$ is equal to

hard

(JEE MAIN-2022)

Let $x_1, x_2, \ldots, x_6$ be the roots of the polynomial equation $x^6+2 x^5+4 x^4+8 x^3+16 x^2+32 x+64=0$. Then,

normal

(KVPY-2017)

Suppose $a, b, c$ are positive integers such that $2^a+4^b+8^c=328$. Then, $\frac{a+2 b+3 c}{a b c}$ is equal to

hard

(KVPY-2015)

The number of solutions of the equation $\log _{(x+1)}\left(2 x^{2}+7 x+5\right)+\log _{(2 x+5)}(x+1)^{2}-4=0, x\,>\,0$, is $....$

Solution

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