What times what equals 27 and adds up too 20?

Answer 1

Let A, B be some numbers

A x B = 27

A + B = 20

Solve for A or B in either equation, than plug what you solved for into the other equation.

A + B = 20

A = 20 - B <--- solve for one of the variables

A x B = 27

(20 - B) x B = 27 <--- plug variable you solved into other equation

20B - B2 - 27 = 0

(-1)(B2-20B+27) = 0 <--- factored out a negative one to make problem easier

B2-20B+27 = 0 <--- cannot factor, need quadratic equation

B = ( -(-20) ± √[(-20)2 - 4(1)(27)] ) / ( 2(1) )

B = 10 ± √(400 - 108)/2

B = 10 ± √(292)/2

B = 10 ± 2√(73)/2

B = 10 ± √73

From here, you could plug B back into one of the equations to solve for A, but as it turns out we already have both numbers.

10 ± √73 = 10 + √73 AND 10 - √73

Check our work:

(10 + √73)(10 - √73) = 100 - 73 = 27 <--- this check works (AB=27)

(10 + √73) + (10 - √73) = 10 + 10 = 20 <--- this check works (A + B = 20)

The answers are 10+√73 and 10-√73