How would you find the points of intersection of the parabolas y equals 4x2 -2x -1 and y equals 3x -2x2 plus 5 by completing the square?

Answer 1

If: y = 4x^2 -2x -1 and y = 3x-2x^2 +5

Then: 4x^2 -2x -1 = 3x -2x^2 +5

Transposing terms: 6x^2 -5x -6 = 0

Dividing all terms by 6: x^2 -5/6 -1 = 0

Completing the square: (x-5/12)^2 = 169/144

Square root both sides: x-5/12 = -13/12 or 13/12

Adding 5/12 to both sides: x = -2/3 or 3/2

Substitution of x into original equations intersections are at: (-2/3, 19/9) and (3/2, 5)

Answer 2

At the points of intersection, 4x2 - 2x - 1 = y = 3x - 2x2 + 5Therefore 6x2 - 5x - 6 = 0

=> x2 - 5x/6 - 1 = 0

=> x2 - 5x/6 = 1

=> (x - 5/12)2 = 1 + (5/12)2 = 1 + 25/144 = 169/144

=> (x - 5/12) = sqrt(169/144) = +/- 13/12

=> x = 5/12 +/- 13/12

=> x = -8/12 or 18/12

=> x = -2/3 or 3/2.

Incidentally, I would never use "completing the squares" to solve a quadratic. Factorising - if you can see the factor easily - or the quadratic formula are far simpler methods.