Answer
Verified
477.9k+ views
Hint: Square both sides to get and substitute $\dfrac{7x-3}{2x+1}=t$. Take care of extraneous roots which will be due to the quantity inside the square root is $\ge 0$ and denominator of each fraction $\ne 0$. Remove those values at which the above conditions are not satisfied.
Complete step-by-step solution -
Let $t=\dfrac{7x-3}{2x+1}$
So, we have
$\begin{align}
& \dfrac{18\left( 7x-3 \right)}{2x+1}=\dfrac{250\sqrt{2x+1}}{3\sqrt{7x-3}} \\
& \Rightarrow \dfrac{18\left( 7x-3 \right)}{2x+1}=\dfrac{250}{3}\sqrt{\dfrac{2x+1}{7x-3}} \\
& \Rightarrow \dfrac{18\left( 7x-3 \right)}{2x+1}=\dfrac{250}{3\sqrt{\dfrac{7x-3}{2x+1}}} \\
& \Rightarrow 18t=\dfrac{250}{3\sqrt{t}} \\
\end{align}$
Cross multiplying, we get
$54{{t}^{\dfrac{3}{2}}}=\dfrac{250}{3}$
Dividing both sides by 54 we get
$\begin{align}
& \dfrac{54{{t}^{\dfrac{3}{2}}}}{54}=\dfrac{250}{3\times 54} \\
& \Rightarrow {{t}^{\dfrac{3}{2}}}=\dfrac{125}{81} \\
& \Rightarrow {{t}^{\dfrac{3}{2}}}={{\left( \dfrac{5}{3} \right)}^{3}} \\
& \Rightarrow t={{\left( \dfrac{5}{3} \right)}^{3\times \dfrac{2}{3}}} \\
& \Rightarrow t={{\left( \dfrac{5}{3} \right)}^{2}} \\
& \Rightarrow t=\dfrac{25}{9} \\
& \Rightarrow \dfrac{7x-3}{2x+1}=\dfrac{25}{9} \\
\end{align}$
Cross multiplying, we get
$9\left( 7x-3 \right)=25\left( 2x+1 \right)$
Simplifying we get
$63x-27=50x+25$
Transposing 27 to RHS and 50x to LHS we get
$63x-50x=25+27$
Simplifying we get
$13x=52$
Dividing both sides by 13, we get
$\dfrac{13x}{13}=\dfrac{52}{13}$
x = 4
Also, we need to check if the terms under square root are non-negative
We have $7x-3=28-3=25\ge 0$ which satisfies the condition
$2x+1=8+1=9\ge 0$ which also satisfies the condition
Also, we need to check that the terms in the denominator are not equal to 0
Since 2x+1 = 9 and 7x – 3 = 25 are both not equal to 0 we have all the conditions satisfied.
Hence x = 4 is the root of the equation $\dfrac{18\left( 7x-3 \right)}{2x+1}=\dfrac{250\sqrt{2x+1}}{3\sqrt{7x-3}}$
Note: [1] Verification: $\dfrac{18\left( 7x-3 \right)}{2x+1}=\dfrac{18\times 25}{9}=50$
$\dfrac{250\sqrt{2x+1}}{3\sqrt{7x-3}}=\dfrac{250\sqrt{9}}{3\sqrt{25}}=\dfrac{250\times 3}{3\times 5}=50$
Hence $\dfrac{18\left( 7x-3 \right)}{2x+1}=\dfrac{250\sqrt{2x+1}}{3\sqrt{7x-3}}$.
Hence it is verified that x = 4 is the root of the given equation.
[2] Substitution of $t=\dfrac{7x-3}{2x+1}$ was the primary step in solving the question. It reduced the effort in solving the question. Substitutions usually help in solving equations faster. e.g. we can solve the question $\left( x+3 \right)\left( x+4 \right)\left( x+5 \right)\left( x+6 \right)=100$ by expanding and forming a biquadratic and solving that biquadratic which would be very difficult or by substituting ${{x}^{2}}+9x+18=t$ in which case we have
$\left( x+3 \right)\left( x+4 \right)\left( x+5 \right)\left( x+6 \right)=\left[ \left( x+3 \right)\left( x+6 \right) \right]\left[ \left( x+5 \right)\left( x+4 \right) \right]$
= $\left( {{x}^{2}}+9x+18 \right)\left( {{x}^{2}}+9x+20 \right)=t\left( t+2 \right)=\text{ }100$
In this case, it is equivalent to solving two quadratic equations and is much easier than solving the biquadratic equation.
Complete step-by-step solution -
Let $t=\dfrac{7x-3}{2x+1}$
So, we have
$\begin{align}
& \dfrac{18\left( 7x-3 \right)}{2x+1}=\dfrac{250\sqrt{2x+1}}{3\sqrt{7x-3}} \\
& \Rightarrow \dfrac{18\left( 7x-3 \right)}{2x+1}=\dfrac{250}{3}\sqrt{\dfrac{2x+1}{7x-3}} \\
& \Rightarrow \dfrac{18\left( 7x-3 \right)}{2x+1}=\dfrac{250}{3\sqrt{\dfrac{7x-3}{2x+1}}} \\
& \Rightarrow 18t=\dfrac{250}{3\sqrt{t}} \\
\end{align}$
Cross multiplying, we get
$54{{t}^{\dfrac{3}{2}}}=\dfrac{250}{3}$
Dividing both sides by 54 we get
$\begin{align}
& \dfrac{54{{t}^{\dfrac{3}{2}}}}{54}=\dfrac{250}{3\times 54} \\
& \Rightarrow {{t}^{\dfrac{3}{2}}}=\dfrac{125}{81} \\
& \Rightarrow {{t}^{\dfrac{3}{2}}}={{\left( \dfrac{5}{3} \right)}^{3}} \\
& \Rightarrow t={{\left( \dfrac{5}{3} \right)}^{3\times \dfrac{2}{3}}} \\
& \Rightarrow t={{\left( \dfrac{5}{3} \right)}^{2}} \\
& \Rightarrow t=\dfrac{25}{9} \\
& \Rightarrow \dfrac{7x-3}{2x+1}=\dfrac{25}{9} \\
\end{align}$
Cross multiplying, we get
$9\left( 7x-3 \right)=25\left( 2x+1 \right)$
Simplifying we get
$63x-27=50x+25$
Transposing 27 to RHS and 50x to LHS we get
$63x-50x=25+27$
Simplifying we get
$13x=52$
Dividing both sides by 13, we get
$\dfrac{13x}{13}=\dfrac{52}{13}$
x = 4
Also, we need to check if the terms under square root are non-negative
We have $7x-3=28-3=25\ge 0$ which satisfies the condition
$2x+1=8+1=9\ge 0$ which also satisfies the condition
Also, we need to check that the terms in the denominator are not equal to 0
Since 2x+1 = 9 and 7x – 3 = 25 are both not equal to 0 we have all the conditions satisfied.
Hence x = 4 is the root of the equation $\dfrac{18\left( 7x-3 \right)}{2x+1}=\dfrac{250\sqrt{2x+1}}{3\sqrt{7x-3}}$
Note: [1] Verification: $\dfrac{18\left( 7x-3 \right)}{2x+1}=\dfrac{18\times 25}{9}=50$
$\dfrac{250\sqrt{2x+1}}{3\sqrt{7x-3}}=\dfrac{250\sqrt{9}}{3\sqrt{25}}=\dfrac{250\times 3}{3\times 5}=50$
Hence $\dfrac{18\left( 7x-3 \right)}{2x+1}=\dfrac{250\sqrt{2x+1}}{3\sqrt{7x-3}}$.
Hence it is verified that x = 4 is the root of the given equation.
[2] Substitution of $t=\dfrac{7x-3}{2x+1}$ was the primary step in solving the question. It reduced the effort in solving the question. Substitutions usually help in solving equations faster. e.g. we can solve the question $\left( x+3 \right)\left( x+4 \right)\left( x+5 \right)\left( x+6 \right)=100$ by expanding and forming a biquadratic and solving that biquadratic which would be very difficult or by substituting ${{x}^{2}}+9x+18=t$ in which case we have
$\left( x+3 \right)\left( x+4 \right)\left( x+5 \right)\left( x+6 \right)=\left[ \left( x+3 \right)\left( x+6 \right) \right]\left[ \left( x+5 \right)\left( x+4 \right) \right]$
= $\left( {{x}^{2}}+9x+18 \right)\left( {{x}^{2}}+9x+20 \right)=t\left( t+2 \right)=\text{ }100$
In this case, it is equivalent to solving two quadratic equations and is much easier than solving the biquadratic equation.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE