Question 122616
Let one of the numbers be represented by x and the other one by y
.
You are told that the sum of the two numbers is 4. This can be written in equation form as:
.
{{{x + y = 4}}}
.
You are also told that the product of the two numbers is 13. This can be written in equation
form as:
.
{{{x*y = 13}}}
.
So you now have two equations, each with the two unknowns and you need to solve them. One way
to get a solution is to use substitution. Solve one of the equations by finding one of the
unknowns in terms of the other and substitute this result into the other equation.
.
Let's begin by solving the product equation for y in terms of x. Divide both sides of the
product equation by x and you get:
.
{{{y = 13/x}}}
.
Then in the sum equation you can substitute {{{13/x}}} for y and you get:
.
{{{x + 13/x = 4}}}
.
Get rid of the denominator by multiplying both sides (all terms) of this equation by x
to get:
.
{{{x^2 + 13 = 4x}}}
.
Get this into standard quadratic form by subtracting 4x from both sides to convert the
equation to:
.
{{{x^2 - 4x + 13 = 0}}}
.
Solve this by using the quadratic formula. Since this equation is of the standard form:
.
{{{ax^2 + bx + c = 0}}}
.
by comparing this standard form term-by-term with with the equation for this problem you
can see that a = 1, b = -4, and c = 13. The quadratic formula says that the solution to
a quadratic equation of the standard form is given by:
.
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
.
So all that you have to do is to substitute your values for a, b, and c into this solution
equation and you have:
.
{{{x = (-(-4) +- sqrt((-4)^2-4*1*13 ))/(2*1) }}}
.
This simplifies to:
.
{{{x = (4 +- sqrt(16-52))/2 =(4+-sqrt(-36))/2}}}
.
But {{{sqrt(-36) = sqrt(36*i^2) = sqrt(36)*sqrt(i^2) = 6*i}}}
.
and substituting this result into the answer simplifies it to:
.
{{{x = (4+-sqrt(-36))/2 = (4 +- 6i)/2 = 2 +- 3i}}}
.
So two possible answers for x are {{{x = 2+3i}}} and {{{x = 2-3i}}}
.
To find the corresponding values of y, return to the sum equation which said that:
.
{{{x + y = 4}}}
.
Substitute {{{x = 2+3i}}} into this equation and you get:
.
{{{2+3i+y = 4}}}
.
Solve for y by subtracting {{{2 + 3i}}} from both sides and you get:
.
{{{y = 4 - 2 - 3i = 2 - 3i}}}
.
This tells you that if {{{x = 2+3i}}} then {{{y = 2 - 3i}}}.
.
Next, solve for the other possible value of x, namely {{{x = 2-3i}}}. Again, start with 
the equation:
.
{{{x + y = 4}}}
.
Substitute {{{x = 2-3i}}} into this equation and you get:
.
{{{2-3i+y = 4}}}
.
Solve for y by subtracting {{{2 - 3i}}} from both sides and you get:
.
{{{y = 4 - 2 + 3i = 2 + 3i}}}
.
This tells you that if {{{x = 2-3i}}} then {{{y = 2 + 3i}}}.
.
Now examining the two sets of answers you see that if one of the numbers (x or y) is {{{2+3i}}} 
then the other number is {{{2-3i}}}. So in either case, that is the answer. One number is
{{{2+3i}}} and the other number is {{{2-3i}}}.
.
Hope this helps you to understand the problem and one way that it can be solved.
.