Answer

Verified

429.9k+ views

Hint: Rather than thinking about dividing we should first consider cross multiplying and see if anything useful comes along. After cross-multiplication, solve using known techniques of algebra to solve for the value of x.

Complete step by step answer:

We have the equation- $\dfrac{10{{x}^{2}}+15x+63}{5{{x}^{2}}-25x+12}=\dfrac{2x+3}{x-5}$

After cross-multiplying we get,

$(x-5)(10{{x}^{2}}+15x+63)=(2x+3)(5{{x}^{2}}-25x+12)$

After multiplying the terms in LHS and RHS and writing them separately we get,

$10{{x}^{3}}+15{{x}^{2}}+63x-50{{x}^{2}}-75x-315=10{{x}^{3}}-50{{x}^{2}}+24x+15{{x}^{2}}-75x+36$

We see that the terms present in both LHS and RHS are- $10{{x}^{3}}$ , $-50{{x}^{2}}$ , $-75x$ , $15{{x}^{2}}$

Therefore the following terms cancels out and we are left with

$63x-315=24x+36$

Subtracting 24x both sides we have

$39x-315=36$

Adding 315 both sides we have,

$\begin{align}

& 39x=351 \\

& \Rightarrow x=\dfrac{351}{39} \\

& \Rightarrow x=9 \\

\end{align}$

Therefore, the above question has only one solution that is x=9.

Hence, the answer is 9.

Note: If we would have tried to solve the question any other way it would have been unnecessarily long. If we would have tried to first divide the terms on LHS and RHS and then proceeded it would also have been rather difficult. Someone may also try to factorise the quadratic equation on the LHS first. As we know factorisation also takes time if we cannot easily see how to split the coefficient of x so that it fits for factorisation. Even after we factorise we again would have to multiply if nothing cancels out. We also saved time where we cancelled out $10{{x}^{3}}$ , $-50{{x}^{2}}$ , $-75x$ , $15{{x}^{2}}$ .

If we had calculated rather than directly cancelling it would have taken twice the time than how we did it. These are some tips to save time when doing these types of questions. The question was set like this to follow this method.

Complete step by step answer:

We have the equation- $\dfrac{10{{x}^{2}}+15x+63}{5{{x}^{2}}-25x+12}=\dfrac{2x+3}{x-5}$

After cross-multiplying we get,

$(x-5)(10{{x}^{2}}+15x+63)=(2x+3)(5{{x}^{2}}-25x+12)$

After multiplying the terms in LHS and RHS and writing them separately we get,

$10{{x}^{3}}+15{{x}^{2}}+63x-50{{x}^{2}}-75x-315=10{{x}^{3}}-50{{x}^{2}}+24x+15{{x}^{2}}-75x+36$

We see that the terms present in both LHS and RHS are- $10{{x}^{3}}$ , $-50{{x}^{2}}$ , $-75x$ , $15{{x}^{2}}$

Therefore the following terms cancels out and we are left with

$63x-315=24x+36$

Subtracting 24x both sides we have

$39x-315=36$

Adding 315 both sides we have,

$\begin{align}

& 39x=351 \\

& \Rightarrow x=\dfrac{351}{39} \\

& \Rightarrow x=9 \\

\end{align}$

Therefore, the above question has only one solution that is x=9.

Hence, the answer is 9.

Note: If we would have tried to solve the question any other way it would have been unnecessarily long. If we would have tried to first divide the terms on LHS and RHS and then proceeded it would also have been rather difficult. Someone may also try to factorise the quadratic equation on the LHS first. As we know factorisation also takes time if we cannot easily see how to split the coefficient of x so that it fits for factorisation. Even after we factorise we again would have to multiply if nothing cancels out. We also saved time where we cancelled out $10{{x}^{3}}$ , $-50{{x}^{2}}$ , $-75x$ , $15{{x}^{2}}$ .

If we had calculated rather than directly cancelling it would have taken twice the time than how we did it. These are some tips to save time when doing these types of questions. The question was set like this to follow this method.

Recently Updated Pages

The branch of science which deals with nature and natural class 10 physics CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Draw a diagram showing the external features of fish class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

How fast is 60 miles per hour in kilometres per ho class 10 maths CBSE