# If vector a = (x^2-7, 27) and vector b = (x+3, -9) are parallel vectors in R^2, determine the value(s) of "x".

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Vector a = (x^2-7, 27) and vector b = (x+3, -9). The two vectors a and b are parallel when the coefficients of the x and y components of the vectors are in the same proportion.

For the same to happen (x^2 - 7)/(x + 3) = 27/-9

=> -9*(x^2 - 7) = 27*(x + 3)

=> x^2 - 7 = -3(x + 3)

=> x^2 - 7 = -3x - 9

=> x^2 + 3x + 2 = 0

=> x^2 + 2x + x + 2 = 0

=> x(x + 2) + 1(x + 2) = 0

=> (x + 1)(x + 2) = 0

=> x = -1 and x = -2

**The variable x = -1 and x = -2**

Let the vector a be:

a = (x^2 - 7)*i + 27*j

Let the vector b be:

b = (x+3)*i - 9*j

For the vectors "a" and "b" to be parallel, then the ratios of corresponding coefficients are equal:

(x^2 - 7)/(x+3) = 27/-9

(x^2 - 7)/(x+3) = -3

We'll cross multiply:

(x^2 - 7) = -3(x+3)

We'll remove the brackets:

x^2 - 7 = -3x - 9

We'll shift all terms to the left side:

x^2 + 3x - 7 + 9 = 0

x^2 + 3x + 2 = 0

We'll apply quadratic formula:

x1 = [-3+sqrt(9-8)]/2

x1 = (-3+1)/2

x1 = -1

x2 = -2

**The variable "x" has two values, when the given vectors are parallel: x1 = -1 and x2 = -2.**