All the possible values of b will be ±22 and ±10. Thus, there are 4 possibilities such that x2 + bx + 21 factors.
We have been the equation as
x2 + bx + 21
Now all the possible factors of 21 are ±1, ±3, ±7 and ±21
We can write the linear factors as (x+)(x+) or (x-)(x-) as the coefficient of x2 is 1 .
The possible linear factors are :
(x + 1)(x + 21) = x2 + 22x + 21
(x + 3)(x + 7) = x2 + 10x +21
(x – 1)(x – 21) = x2 – 22x + 20
(x – 3)(x – 7) = x2 – 10x + 20
Hence all the possible values of b will be ±22 and ±10. There are 4 possibilities.
Learn more about factorization here : brainly.com/question/25829061
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Start by, 60/30= then 2*20= 40 so 40 bottles for 20 runners.
Step-by-step explanation:
Let the two numbers be x and y.
From equations (1) & (2)
![(2-y)y=-8\\\therefore \: 2y - y^2 = - 8\\\\\therefore \:y^2 - 2y-8=0\\\\\therefore \:y^2 - 4y+2y-8=0\\\\\therefore \:y(y - 4)+2(y-4)=0\\\\\therefore \:(y+2)(y - 4)=0\\\\\therefore \:(y+2)=0\:\: or\:\:(y - 4)=0\\\\\therefore \:y=-2\:\: or\:\:y =4\\\\ When \:\: y = - 2\implies x = 4\\\\When \:\: y = 4\implies x = - 2\\](https://tex.z-dn.net/?f=%20%282-y%29y%3D-8%5C%5C%3C%2Fp%3E%3Cp%3E%5Ctherefore%20%5C%3A%202y%20-%20y%5E2%20%3D%20-%208%5C%5C%5C%5C%3C%2Fp%3E%3Cp%3E%5Ctherefore%20%5C%3Ay%5E2%20-%202y-8%3D0%5C%5C%5C%5C%3C%2Fp%3E%3Cp%3E%5Ctherefore%20%5C%3Ay%5E2%20-%204y%2B2y-8%3D0%5C%5C%5C%5C%3C%2Fp%3E%3Cp%3E%5Ctherefore%20%5C%3Ay%28y%20-%204%29%2B2%28y-4%29%3D0%5C%5C%5C%5C%3C%2Fp%3E%3Cp%3E%5Ctherefore%20%5C%3A%28y%2B2%29%28y%20-%204%29%3D0%5C%5C%5C%5C%3C%2Fp%3E%3Cp%3E%5Ctherefore%20%5C%3A%28y%2B2%29%3D0%5C%3A%5C%3A%20or%5C%3A%5C%3A%28y%20-%204%29%3D0%5C%5C%5C%5C%3C%2Fp%3E%3Cp%3E%5Ctherefore%20%5C%3Ay%3D-2%5C%3A%5C%3A%20or%5C%3A%5C%3Ay%20%3D4%5C%5C%5C%5C%20%20%20%20%3C%2Fp%3E%3Cp%3EWhen%20%5C%3A%5C%3A%20y%20%3D%20-%202%5Cimplies%20x%20%3D%204%5C%5C%5C%5C%3C%2Fp%3E%3Cp%3EWhen%20%5C%3A%5C%3A%20y%20%3D%204%5Cimplies%20x%20%3D%20-%202%5C%5C%3C%2Fp%3E%3Cp%3E)
Thus, the two numbers are (4, - 2) or (-2, 4).