Verify if the curves y=x^2+x+1 and y=-x^2-2x+6 have common points?

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have the curves y=x^2+x+1 and y=-x^2-2x+6. At the common point the x and y coordinates are the same.

To find the common point equate x^2+x+1 and -x^2-2x+6. If there is a real solution for x, the curves have a common point.

x^2+x+1 = -x^2-2x+6

=> 2x^2 + 3x - 5 = 0

=> 2x^2 + 5x - 2x - 5 = 0

=> x(2x + 5) - 1(2x + 5) = 0

=> (x - 1)(2x + 5) = 0

=> x = 1 and x = -5/2

y = 3 and y = 19/4

The common points are (1, 3) and (-5/2, 19/4)

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll equate the given expressions:

x^2+x+1 =-x^2-2x+6

We'll shift all terms to one side:

x^2 + x^2 + x + 2x + 1 - 6 = 0

We'll combine like terms:

2x^2 + 3x - 5 = 0

We'll apply quadratic fromula:

x1 = [-3+sqrt(9+40)]/4

x1 = (-3+7)/4

x1 = 1

x2 = (-3-7)/4

x2 = -10/4

x2 = -5/2

We'll get y1 coordinate, when x1 = 1:

y1 = 1+1+1=3 (we notice that we'll get the same value if we'll replace x by 1 in the 2nd equation).

We'll get y2 coordinate, when x2 = -5/2:

y2 = 25/4 - 5/2 + 1

y2 = 15/4 + 1

y2 = 19/4

The intercepting points of the given curves are: (1 ; 3) and (-5/2 ; 19/4).

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