Answer

Verified

410.7k+ views

**Hint:**First of all we will take the LCM of the denominator of the left-hand side. And we will solve it. After that we will do the cross multiplication then we will get an equation and will keep the equation to one side and equate it with the zero and we will solve for $x$.

**Complete step-by-step answer:**We have

\[\dfrac{1}{{x + 1}} + \dfrac{3}{{5x + 1}} = \dfrac{5}{{x + 4}}\]

Now taking the LCM and solving for it, we get

$ \Rightarrow \dfrac{{5x + 1 + 3x + 1}}{{\left( {x + 1} \right)\left( {5x + 1} \right)}} = \dfrac{5}{{x + 4}}$

On solving the numerator and denominator, we get

$ \Rightarrow \dfrac{{8x + 4}}{{5{x^2} + 6x + 1}} = \dfrac{5}{{x + 4}}$

Now on expanding the equation and doing the cross-multiplication, we get

$ \Rightarrow \left( {8x + 4} \right)\left( {x + 4} \right) = 5\left( {5{x^2} + 6x + 1} \right)$

On solving the multiplication part in both the side, we get

$ \Rightarrow 8{x^2} + 36x + 16 = 25{x^2} + 30x + 5$

On equating the equation and taking all terms on one side, we get

$ \Rightarrow 17{x^2} - 6x - 11 = 0$

Now we will solve the above equation by using the method called factor theorem.

So by splitting the middle term, we will solve the equation, we get

$ \Rightarrow 17{x^2} - 17x + 11x - 11 = 0$

Now taking the common in LHS, we get

$ \Rightarrow 17x\left( {x - 1} \right) + 11\left( {x - 1} \right) = 0$

And it can be written as

$ \Rightarrow \left( {17x + 11} \right)\left( {x - 1} \right) = 0$

Now on equating both the term with zero, we get

$ \Rightarrow x = \left( {\dfrac{{ - 11}}{{17}}} \right)$ Or $x = 1$.

**Therefore, $x = \left( {\dfrac{{ - 11}}{{17}}} \right)$ or $x = 1$ will be the value for the $x$.**

**Additional information:**

Factor theorem is utilized when considering the polynomials. It is a hypothesis that connects variables and zeros of the polynomial. The factor theorem is regularly utilized for figuring a polynomial and finding the foundations of the polynomial. It is an uncommon instance of a polynomial leftover portion hypothesis.

**Note:**This question can also be solved in different ways. Like we can also use the quadratic formula to solve it. $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$. By using this we can solve any quadratic equations easily. One another method is by using the graphical method.

Recently Updated Pages

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

Advantages and disadvantages of science

10 examples of friction in our daily life

Trending doubts

Which are the Top 10 Largest Countries of the World?

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

Who was the first to raise the slogan Inquilab Zindabad class 8 social science CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

One cusec is equal to how many liters class 8 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

A resolution declaring Purna Swaraj was passed in the class 8 social science CBSE