Question

What two numbers multiply to -40 and add to 10

Answer 1

Answer:

5 + sqrt(65), 5 - sqrt(65)

Step-by-step explanation:

x + y = 10

x×y = -40

so, two equations with 2 variables. using one equation to express one variable by the second. and then using the second equation to solve and calculate the second variable. and then by knowing the second variable we calculate the first.

=> x = 10 - y

=> (10-y)×y = -40

=> -1×(10-y)×y = 40

=> (y-10)×y = 40

=> y^2 - 10×y - 40 = 0

remember the solution for a quadratic equation

a×x^2 + b×x + c

is

x = (-b ± sqrt(b^2 - 4×a×c)/(2×a)

in our case

a = 1

b = -10

c = -40

and we used "y".

so,

y = (--10 ± sqrt((-10)^2 - 4×1×(-40))/(2×1) =

= (10 ± sqrt(100 + 160))/2 = (10 ± sqrt(260))/2 =

= (10 ± sqrt (4×65))/2 = 10/2 ± 2×sqrt(65)/2 =

= 5 ± sqrt(65)

=>

for y = 5 + sqrt(65), x = 10 - 5 - sqrt(65) = 5 - sqrt(65)

for y = 5 - sqrt(65), x = 10 - 5 + sqrt(65) = 5 + sqrt(65)