Eight more then the square of a number is the same as six times the number. What is the number?

Question

that's the whole question so....

Answer 1

The key to verbal problem is to translate the words into a mathematical expression.

"Eight more then the square of a number":    the number is unknown thus it is "x".  The square of the number is x^2.

Eight more is 8+x^2.

So 8+ x^2 "is the same as  six times the number".  Our number is "x" and six times of the number is 6x.

Therefore our equation is:   8 + x^2 = 6x 

Subtracting 6x from both sides of the equation is:   8 + x^2 - 6x = 0

Rearranging it                                                      x^2 - 6x + 8 =0

This can be solved by the quadratic equation:  ( -b +- (b^2 - 4ac)1/2)/ 2a for ax^2 + bx + c =0

 (6+-(6*6 - 4*1*8)1/2/2*1  

(6+-(36-32)1/2)/2

(6+-(4)1/2)/2

The two solutions are:  

(6+2)/2   and (6-2)/2

4 and 2

 

Alternatively one can recognize that the equation x^2 - 6x + 8 = (x-4)(x-2) = 0  so x = 4 or x = 2

 

Therefore there are two numbers that satisfies this question 4 and 2.