Question

Determine the points of intersection of the line y = -2x + 7 and the parabola y = 2x2 + 3x - 5.

Answer 1

The points of intersection lie at (1.5, 4) and (-4, 15)

Explanation:

We need to find where both functions are equal, so we will equate them

-2x + 7 = 2x² + 3x - 5 move everything to one side

0 = 2x² + 5x - 12 use the quadratic formula

x = (-5 ± √(5² - 4(2 * -12))) / 2(2)

x = (-5 ± √(121)) / 4

x = (-5 + 11) / 4; x = (-5 - 11) /4

x = 1.5 x = -4

our x values then are 1.5 and -4.

y = -2x + 7

y = -2(1.5) +7

y = 4

(1.5, 4)

y = -2(-4) +7

y = 15

(-4, 15)

y = 2(1.5)² + 3(1.5) - 5

y = 4

(1.5, 4)

y = 2(-4)² + 3(-4) -5

y = 15

(-4, 15)

Answer 2

(-4, 15), (1.5, 4)

Explanation:

This is easily solved using a graphing calculator. (See attached.)

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Algebraically, the solution is found by equating the expressions for y:

... -2x +7 = y = 2x² +3x -5

... 2x² +5x -12 = 0 . . . . rearranged to standard form

... (2x -3)(x +4) = 0 . . . . factor

The x-values of the points of intersection are those that make these factors be zero:

... 2x-3 = 0 ⇒ x = 3/2

... x+4 = 0 ⇒ x = -4

The linear equation is the easiest to use to find the corresponding y-values.

... for x = 3/2, y = -2(3/2) +7 = 4

... for x = -4, y = -2(-4) +7 = 15

The points of intersection are (-4, 15) and (3/2, 4).