Question

What two numbers add to 4 and when multiplied together equal 5

Answer 1

`\large\text{no real solution}\\\\\text{complex solution}`

`\large\boxed{x=2+i\ and\ y=2-i\ or\ x=2-i\ and\ y=2+i}`

Explanation:

`x,\ y-\text{the numbers}\\\\x+y=4\\xy=5\\\\x+y=4\qquad\text{subtract y from both sides}\\x=4-y\\\\\text{substitute it to the second equation}\\\\(4-y)y=5\qquad\text{use distributive property}\\\\4y-y^2=5\\\\-y^2+4y=5\qquad\text{change the signs}\\\\y^2-4y=-5\\\\y^2-2(y)(2)=-5\qquad\text{add}\ 2^2\ \text{to both sides}\\\\y^2-2(y)(2)+2^2=-5+2^2\qquad\text{use}\ (a-b)^2=a^2-2ab+b^2\\\\(y-2)^2=-5+4\\\\(y-2)^2=-1<0\to\boxed{no\ real\ solution}`

`\text{If you need the complex solution, then}\\\\(y-2)^2=-1\to y-2=\pm√(-1)\\\\y-2=-i\ \vee\ y-2=i\qquad\text{add 2 to both sides}\\\\\boxed{y=2-i\ \vee\ y=2+i}\\\\\text{Put the values of y to the first equation:}\\\\x=4-(2-i)\ \vee\ x=4-(2+i)\\\\x=4-2+i\ \vee\ x=4-2-i\\\\\boxed{x=2+i\ \vee\ x=2-i}`