Question 95514
Call the original unknown number x
.
Then the square of the original number is x^2
.
If you add the square of the number to the original number, the sum is 90. In equation form
this is:
.
x^2 + x = 90
.
Subtract 90 from both sides of this equation so that the right side becomes zero. After this
subtraction of 90 from both sides the equation becomes:
.
x^2 + x - 90 = 0
.
The left side of this equation can be factored to give:
.
(x + 10)(x - 9) = 0
.
(If you multiply the two factors on the left side, the product will be x^2 + x - 90.)
.
Notice now that this equation will be true if either of the two factors on the left side
is equal to zero.  This is because zero times the other factor will be zero and that will
equal the right side.
.
So the equation will be true if either x + 10 equals zero, or if x - 9 equals zero.
.
Solving these two equations will result in two values for x that will make quadratic
equation true.
.
Solve x + 10 = 0 by subtracting 10 from both sides to get x = -10.  
.
Solve x - 9 = 0 by adding 9 to both sides to get x = +9
.
Therefore, the greater value of x is +9 and the lesser value is -10. (Greater value means
that the number is further to the right on the number line. Since +9 is further to the
right than -10, it is the greater value.)
.
Check by going back to the original problem.  If x is -10 then x^2 is +100 and the sum of
these two is +90. That works.
.
Next if x is +9 then x^2 is +81 and the sum of these to is also +90. So the two answers work.
.
Hope this helps you to understand the problem and how to work it to get a solution.
.