★ Concept :-

Here the concept of QUADRATIC Equation has been used. We see that we are given an equation and we need to find it's roots. One way is given using hint. We can use it. If there's any problem in using that, we can SIMPLY solve this question as it is. The easiest method to do such questions is using Splitting the Middle Term. Here we shall look out at both steps that is using the hint given in qúestion and directly using the qúestion.

Let's do it !!

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★ Solution :-

Given,

» 18x² + 21x - 49 = 0

• Method I ::

In this method we shall assume the hint given in the qúestion. It's given that,

→ t = 3x

Squaring both sides, we get

→ t² = 9x²

Then,

→ 18x² = 2t²

→ 21x = 7t

Now we can apply this in the given equation.

>> 2t² + 7t - 49 = 0

On splitting the middle term, we get

>> 2t² + 14t - 7t - 49 = 0

>> 2t(t + 7) - 7(t + 7) = 0

On grouping, we get

>> (2t - 7)(t + 7) = 0

Here either (2t - 7) = 0 or (t + 7) = 0.

Then we get,

>>> 2t - 7 = 0 or t + 7 = 0

>>> 2t = 7 or t = - 7

>>> t = 7/2 or t = -7

We know that,

→ t = 3x

→ x = t/3

Since we have TWO values of t, so we can apply them simultaneously and get,

→ x = 7/6 , -7/3

These are the roots of the given equation.

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• Method II ::

In this method, we shall directly solve the equation using the Method of Splitting the Middle Term.

» 18x² + 21x - 49 = 0

This can be written as,

>> 18x² + 42x - 21x - 49 = 0

Now taking the terms in common, we get

>> 6x(3x + 7) - 7(3x - 7) = 0

On grouping, we get

>> (6x - 7)(3x + 7) = 0

Here either (6x - 7) = 0 or (3x + 7) = 0. So,

>> 6x - 7 = 0 or 3x + 7 = 0

>> 6x = 7 or 3x = -7

>> x = 7/6 or x = -7/3

Clearly, this solution MATCHES with the one we got above. So our answer is correct.