What 2 numbers multiply to sixty and also add up to four?

Answer 1

This is the same as a quadratic, however, I'll show you all the steps to get there. Once you get more proficient at math, you can jump straight to the quadratic step. This answer will have a complex solution, not 20 as one of the terms as the previous posted stated.

Let n, m be any number

n x m = 60

n + m = 4

Two equations, two variables, this is solvable.

First solve one of your two equations for one of the variables. I'll solve the equation with addition (it's easier) for n.

n + m = 4

n = 4 - m

Now plug the n into your first equation.

n x m = 60

(4 - m) x m = 60

4m - m2= 60

4m - m2 - 60 = 0 (do this so you can use the zero factor theorem)

Factor out a negative one, because it makes this problem easier:

-1(m2 - 4m + 60) = 0

-1 does not equal 0, so you can drop it

m2- 4m + 60 = 0 <-- this is the quadratic

You can either try to factor it (to solve), or use the quadratic equation, where:

am2+ bm + c = 0

m = [ -b+√ (b2- 4ac) ] / ( 2a )

m = [ -(-4)+√ ((-4)2- 4(1)(60)) ] / ( 2(1) )

m = [ 4+ √ ( 16- 240 ) ] / ( 2 ) <-- a negative square root is going to give an imaginary number

m = [ 4+ √ ( 224) i ] / ( 2 ) <-- √-1 = i, so we factored it out

m = [ 4+ 4√14 i ] / ( 2 ) <-- √224 simplifies to 4√14

m = 2+ 2√14 i <-- divided both terms by 2

You will get two values of m, one of the terms may not work.

m = 2+2√14 i

m = 2-2√14 i

Remember, n = 4 - m, and we'll use the first m term:

n = 4 - ( 2+2√14 i )

n = 2+2√14 i

Check your work:

nxm = 60

( 2+2√14 i )( 2-2√14 i) <-- i^2 = -1

4 - 4√14 i + 4√14 i - 4(14)(-1)

4 + 4(14) = 4 + 56 = 60

n + m = 4

( 2+2√14 i ) + ( 2-2√14 i)

the 2√14 i cancel out and you are left with

2 + 2 = 4

Therefore the answer is:

2+2√14 i

and

2-2√14 i