Question

What two numbers multiply to -294 and add up to 7?

Answer 1

Answer: -14 and 21

Explanation:

Let two numbers will be x and y.

Hence,

`\displaystyle\\\left \{ {{x+y=7} \atop {x*y=-294}} \right.\\\\\left \{ {{y=7-x} \atop {x*(7-x)=-294}} \right. \\\\\left \{ {{y=7-x} \atop {7x-x^2=-294}} \right. \\\\\left \{ {{y=7-x} \atop {7x-x^2+x^2=-294+x^2}} \right. \\\\\left \{ {{y=7-x} \atop {7x=x^2-294}} \right. \\\\\left \{ {{y=7-x} \atop {7x-7x=x^2-294-7x}} \right. \\\\\left \{ {{y=7-x} \atop {0=x^2-7x-294}} \right. \\\\x^2-7x-294=0\\D=(-7)^2-4*1*(-294)\\D=49+1176\\D=1225\\√(D)=√(1225) \\√(D)=35\\\\`

`\displaystyle\\ x=(-(-7)б35)/(2) \\\\x=(7б35)/(2) \\\\x=-14\\y=7-(-14)\\y=7+14\\y=21\\\\x=21\\y=7-21\\y=-14`

Answer 2

The two numbers 21 and -14 multiply to -294 and add up to 7

Explanation:

We have to solve the system:

a + b = 7

ab = -294

Then

b = 7 - a

a(7 - a) = -294 (substitute b by 7 - a)

Then

b = 7 - a

7a - a² = -294

Then

b = 7 - a

a² - 7a - 294 = 0

Then

b = -294 - a

a² + 294a + 7 = 0

Then

`a=\frac{7+ \sqrt{7^(2)+4 * 294} }{2} =(7+ √(1225) )/(2) = =(7+35)/(2)=(42)/(2) =21`

Or

`a=\frac{7- \sqrt{7^(2)+4 * 294} }{2} =(7- √(1225) )/(2) = =(7-35)/(2)=(-28)/(2) =-14`

Then

b = 7 - a = 7 - 21 = -14

Or

b = 7 - a = 7 - (-14) = 21