Prove that the vertex of parabola y=x^2+5x+1 is in third quadrant.

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To prove that the coordinates of a point are located in the 3rd quadrant of Cartesian coordinate system, we'll have demonstrate that both their values are negative.

We'll calculate the vertex of the parabola using the formula:

V(-b/2a ; -`Delta` /4a),

`Delta` = b^2 - 4ac

a,b,c are the coefficients of quadratic:

a=1, b=5 and c=1.

`Delta` = 25 - 4 = 21

-b/2a = -5/2

-`Delta` /4a = -21/4

Since the x and y coordinates of the vertex of parabola are negative, the vertex V(-5/2 ; -21/4) is located in the 3rd quadrant of Cartesian coordinate system.

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