Question 182230
Let x and y be the two numbers, then:
1) {{{x+y = 8}}} and
2) {{{x*y = 2}}} Rewrite equation 1) as: {{{x = 8-y}}} and substitute into equation 2).
2a) {{{(8-y)*y = 8}}} Simplify.
2a) {{{8y-y^2 = 8}}} Rewrite this in standard quadratic form.
{{{y^2-8y+8 = 0}}} Solve by the quadratic formula:{{{ y= (-b+-sqrt(b^2-4ac))/2a}}} where: a = 1, b = -8, and c = 2.
{{{y = (-(-8)+-sqrt((-8)^2-4(1)(2)))/2(1)}}} Evaluate.
{{{y = (8+-sqrt(64-8))/2}}}
{{{y = (8+-sqrt(56))/2}}}
{{{highlight(y = 4+sqrt(14))}}} or {{{highlight(y = 4-sqrt(14))}}} These are the two real numbers.
Check:
{{{x+y = 8}}} Substitute {{{x = 4+sqrt(14)}}} and {{{y = 4-sqrt(14)}}}
{{{(4+sqrt(14))+(4-sqrt(14)) = 8+sqrt(14)-sqrt(14)}}} = 8
{{{x*y = 2}}} Substitute for x and y as above.
{{{(4+sqrt(14))*(4-sqrt(14)) = 16-(sqrt(14))*(sqrt(14))}}} = {{{16-14 = 2}}}