Thrice the square of a natural number decreased by four times the number is equal to 50 more than the number, the number is -

Solution(By Examveda Team)

Let the number be x
According to question
$$\eqalign{ & \left( {3 \times {x^2}} \right){\text{ - }}\left( {4 \times x} \right) = 50 + x.....(i) \cr & \Rightarrow 3{x^2} - 4x = 50 + x \cr & \Rightarrow 3{x^2} - 5x - 50 = 0 \cr & \Rightarrow 3{x^2} - 15x + 10x - 50 = 0 \cr & \Rightarrow 3{x^{}}(x - 5) + 10(x - 5) = 0 \cr & \Rightarrow (x - 5)(3x + 10) = 0 \cr & x = 5\,\,or\,\, - \frac{{10}}{3} \cr} $$
Since the natural number is x = 5

Shortcut method :
$$ \Rightarrow 3{x^2} - 4x = 50 + x.....(i)$$
Now put the value of x from option (b)
$$\eqalign{ & x = 5 \cr & 3 \times {(5)^2} - 4 \times 5 = 50 + 5 \cr & 75 - 20 = 55 \cr & 55 = 55 \cr} $$
LHS = RHS (it satisfies the conditions)
$${\text{so, x = 5}}$$