Prove that the line y=2x-1 does not intersect the parabola y=x^2+x+1 .

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tonys538's profile pic

tonys538 | Student, Undergraduate | (Level 1) Valedictorian

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If two curves intersect, there exists a real solution when the set of equations representing the curves are solved.

Solving the equations y=2x-1 and y=x^2+x+1 gives:

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

x^2 - 2x + 2 = 0

A quadratic equation ax^2 + bx + c = 0 has real roots if `b^2 - 4ac >= 0`

Here, a = 1, b = -2 and c = 2

b^2 - 4ac = 4 - 8 = -4

As a result, there is no real solution for the equation of the line and the equation of the parabola.

This proves that the line y=2x-1 and the parabola y=x^2+x+1 do not intersect.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

To prove that the line and parabola are not intercepting each other, we'll have to prove that the below equation has no real solutions.

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

We'll use the symmetrical propert and we'll move all terms to one side:

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

We'll combine like terms:

x^2 - x + 2 = 0

We'll apply quadratic formula:

x1 = [1+sqrt(1-8)]/2

x1 = 1/2 + i*sqrt7/2

x2 = 1/2 - i*sqrt7/2

As we can notice, the roots of the equation are imaginary numbers, therefore there are no real numbers to satisfy both equations.

Since the equation 2x - 1 = x^2 + x + 1 has no real solutions, the line and parabola are not intercepting each other.

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