Question

a number exceeds 4 times its reciprocal by 3

Answer 1

If so: 

Let the number be x. 

The reciprocal of a number is simply 1/the number so the reciprocal of x is 1/x. 
4 times the reciprocal is 4(1/x) which is (4/x) 
But the number exceeds it by 3, or in other words it is 3 greater than it 

So the equation you end up with is: 

(4/x) + 3 = x 
4 + 3x = x^2 ...... multiply both sides by x 
x^2 - 3x - 4 = 0 
(x - 4)(x + 1) = 0 .... factorize 

so x = 4 or x = -1 

we have two values because we multiplied by x earlier. But one of these values is false. 
So test them yourself by substituting each value for x into the first equation. 

You will see that the right answer is x= 4

Answer 2

let,
the required  number = x
reciprocal of required number = 1/x
4 times the reciprocal = 4(1/x)
according to the question,
x = 4(1/x) +3
x = 4/x +3
x =( 4 + 3x) / x          by taking the lcm
x² = 4 + 3x
x²- 3x -4 =0              by cross multiplying
x²-4x +x - 4 = 0
x(x-4)+1(x-4) = 0
(x+1)(x-4)      = 0
x = -1, x = 4
the required no.= x = -1 and 4