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olga55 [171]
3 years ago
10

What two numbers multiply to 44 and add up to 12?

Mathematics
2 answers:
djverab [1.8K]3 years ago
8 0
This pattern of question is always coming up. Since we can't easily guess, then let us set up simultaneous equation for the statements.

let the two numbers be x and y.

Multiply to 44.      x*y = 44 ..........(a)

Add up to 12.      x + y = 12 .........(b)

From (b)

y = 12 - x .......(c)

Substitute (c) into (a)

x*y = 44

x*(12 - x) = 44   

12x - x² = 44

-x² + 12x = 44

-x² + 12x - 44 = 0.       

Multiply both sides by -1

-1(-x² + 12x - 44) = -1*0

x² - 12x + 44 = 0.   

This does not look factorizable, so let us just use quadratic formula

comparing to ax² + bx + c = 0, x² - 12x + 44 = 0,  a = 1, b = -12, c = 44 

x = (-b + √(b² - 4ac)) /2a   or (-b - √(b² - 4ac)) /2a


x = (-(-12) + √((-12)² - 4*1*44) )/ (2*1)    

x = (12 + √(144 - 176) )/ 2

x = (12 + √-32 )/ 2

√-32 = √(-1 *32) = √-1 * √32 = i * √(16 *2) = i*√16 *√2 = i*4*√2 = 4i√2

Where i is a complex number.  Note the equation has two values. We shall include the second, that has negative sign before the square root.

x = (12 + √-32 )/ 2      or     (12 - √-32 )/ 2   

x = (12 + 4i√2 )/ 2              (12 - 4i√2 )/ 2 

x = 12/2 + (4i√2)/2                12/2 - (4i√2)/2

x = 6 + 2i√2            or         6 - 2i√2

Recall equation (c):

y = 12 - x, When x = 6 + 2i√2,  y = 12 - (6 + 2i√2) = 12 - 6 - 2i√2 = 6 - 2i√2

When x = 6 - 2i√2,  y = 12 - (6 - 2i√2) = 12 - 6 + 2i√2 = 6 + 2i√2


x = 6 + 2i√2,  y = 6 - 2i√2

x = 6 - 2i√2,  y = 6 + 2i√2

Therefore the two numbers that multiply to 44 and add up to 12 are:

6 + 2i√2 and 6 - 2i√2
guapka [62]3 years ago
5 0
xy=44\\
x+y=12\\\\
xy=44\\
x=12-y\\\\
(12-y)y=44\\
12y-y^2=44\\
y^2-12y+44=0\\
y^2-12y+36+8=0\\
(y-6)^2=-8


No solutions in real numbers.

In complex numbers:
(y-6)^2=-8\\
y-6=-\sqrt{-8} \vee y-6=\sqrt{-8}\\
y=6-2\sqrt2 i \vee y=6+2\sqrt2 i\\\\
x=12-(6-2\sqrt2i) \vee x=12-(6+2\sqrt2i)\\
x=12-6+2\sqrt2i \vee x=12-6-2\sqrt2i\\
x=6+2\sqrt2i \vee x=6-2\sqrt2i

These numbers are 6-2\sqrt2i and 6+2\sqrt2i.
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