What 2 numbers add up to -45 and multiply to -196?

Answer 1

1) x + y = -45

2) xy = -196

solve for x in equation 2 by dividing both sides by y

x= -196/y

plug this value for x into the first equation

-196/y + y = -45

subtract y from both sides

-196/y = -45-y

multiply both sides by -y

196 =45y + y2

subtract 196 from both sides

0 = y2 + 45y - 196

factor

0 = (y + 49)(y - 4)

set each factor equal to 0

y + 49 = 0, y = -49

y - 4 = 0, y = 4

y = -49 or 4

plug these values back into equation 1

x + 4 = -45, x = -49

x + -49 = -45, x = 4

so the two numbers are 4 and -49

As two numbers are required that sum to -45 and multiply to -196, start off by working out the prime factorization of 196:

196 = 2 x 2 x 7 x 7.

Using this, pairs of factors of 196 can be calculated:

1 x 196

2 x 98

4 x 49

7 x 28

14 x 14

As -196 is required, one factor must be positive and the other negative which means when they are summed, really their difference is calculated:

1 x 196 - difference = 195

2 x 98 - difference = 96

4 x 49 - difference = 45

7 x 28 - difference = 21

14 x 14 - difference = 0

A difference of 45 means the factors are 4 and 49, but as -45 is required, the larger factor must be negative, meaning the required numbers are 4 and -49

If you are in fact trying to factor a quadratic equation x2 - 45x - 196 = 0, you could also use the quadratic formula [-b + or - the square root of (b2-4ac)] / 2a

so [-(-45) + or - the square root of ((-45)2 - 4(1)(-196)] / 2(1)

[45 + or - square root of (2809)] / 2

[45 + or - 53] / 2

45 + 53 = 98, divide by 2 = 49

45 - 53 = -8, divide by 2 = -4

so the roots are (x - 49)(x + 4) so x = {49, -4}