Anonymous
Anonymous asked in Science & MathematicsMathematics · 1 month ago

# What 2 numbers are multiplied to get -24 but sums up to +6?

help

Update:

Nevermind

Relevance
• a=3- sqrt(33)

b=3+sqrt(33)

• xy = -24...(1).;

x+y = 6 ...(2).;

(1)---> y = -(24/x);

Then (2)---> x = 6+(24/x);

Then x^2 = 6x +24, ie., x^2-6x = 24, ie., x^2-6x+9 = 33, ie., (x-3)^3 = (rt33)^2, ie.,;

x = 3(+/-)rt33;

corresponding y = 3(-/+)rt33;

• The numbers needed are: 3 -square root of 33 and 3 +square root of 33

• x + y = 6, and let

x – y = 2t, so then

x = 3 + t, and

y = 3 - t

We can find t using

4t^2 = (x – y)^2 = (x + y)^2 – 4xy = 36 + 4*24 = 132

t^2 = 33, symmetric results for x and y, so select one, say

x = 3 + √33

y = 3 - √33

Check: (3 + √33)(3 - √33) = 9 – 33 = -24 as required.

• Let the numbers to 'm' & 'n'

Hence

mn = -24

m + n = 6

m = -24/n

Substitute

-24/n + n = 6

Multiply through by 'n'

-24 + n^2 = 6n

n^2 - 6n - 24 = 0

It is now in Quadratic form

Complete the Square

(n - 3)^2 - ( 3)^2 = 24

( n- 3)^2 - 9 = 24

(n - 3)^2 = 33

Square root both sides

n - 3 = +/-sqrt(33)

n = 3 +/-sqrt(33)

n = 3 +/- 5.7445....

n = - 8 .7445.... or - 2.7445....

Hence m = -24/(-8.7445... ) or -24/-2.7445,]...

m = 2.7445... or 8,7445...

So as pairs of numbers (-8.7445... , 2.7445..) & ( 8.7445... , -2.7445..)

• Two numbers that multiply to -24 and adds to 6:

xy = -24 and x + y = 6

x + y = 6

x = 6 - y

xy = -24

(6 - y)y = -24

6y - y² = -24

-y² + 6y = -24

y² - 6y = 24

y² - 6y + 9 = 24 + 9

(y - 3)² = 33

y - 3 = ± √33

y = 3 ± √33

We have two y's which will give us two x's, but they will be the same two numbers.

3 - √33 and 3 + √33

Testing:

3 - √33 + 3 + √33

3 + 3

6 <-- true

(3 - √33)(3 + √33)

9 + 3√33 - 3√33 - 33

9 - 33

-24 <-- true