Question

Find the coordinates of the points a and b where the line y = 2x − 1 cuts the curve y = x2 − 4.

Answer 1

(3,5) and (-1,-3)

Explanation:

2x - 1 = x² - 4

2x = x² - 3

0 = x² - 2x - 3

0 = (x+1)(x-3)

x1 = 3, x2 = -1

y = 2(3) - 1

y = 6 - 1

y = 5

(3,5)

y = 2(-1) - 1

y = -2 - 1

y = -3

(-1,-3)

Therefore, the coordinates are (3,5) and (-1,-3)

Answer 2

(-1, -3) and (3, 5)

Explanation:

I find a graphing calculator to be a very useful tool for finding solutions to problems like this. The attachment shows the points of intersection to be (-1, -3) and (3, 5).

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You can find these points algebraically by substituting for y in either equation using the expression provided by the other equation.

x^2 -4 = y = 2x -1

x^2 -2x +1 = 4 . . . . . . . add 5-2x to make a perfect square trinomial

(x -1)^2 = 4 . . . . . . . . . show as a square

x -1 = ±√4 = ±2 . . . . . take the square root

x = 1 ±2 = {-1, 3} . . . . add 1, show the separate solutions

y = 2x -1 = 2{-1, 3} -1 = {-2, 6} -1 = {-3, 5} . . . . find the corresponding y-values

The points of intersection are (-1, -3) and (3, 5).