Question

The line L has equation
`y + 2x = 12`

and the curve C has equation

`y = {x}^(2) - 4x + 9`
Q1) show that the x coordinates of the points of intersection of L and C satisfy the equation

`{x}^(2) - 2x - 3 = 0`

Q2) hence find the coordinates of the points of intersection of L and C​

Answer 1

see explanation

Explanation:

Q1

given line L has equation

y + 2x = 12 ( subtract 2x from both sides )

y = 12 - 2x → (1)

given curve C has equation

y = x² - 4x + 9 → (2)

substitute y = x² - 4x + 9 into (1)

x² - 4x + 9 = 12 - 2x ( add 2x to both sides )

x² - 2x + 9 = 12 ( subtract 12 from both sides )

x² - 2x - 3 = 0 ← equation for x- coordinates

Q2

to find the x- coordinates of the points of intersection, solve the equation

x² - 2x - 3 = 0 ← in standard form

(x - 3)(x + 1) = 0 ← in factored form

equate each factor to zero and solve for x

x - 3 = 0 ⇒ x = 3

x + 1 = 0 ⇒ x = - 1

substitute these values into (1) for corresponding values of y

x = 3 : y = 12 - 2(3) = 12 - 6 = 6 ⇒ (3, 6 )

x = - 1 : y = 12 - 2(- 1) = 12 + 2 = 14 ⇒ (- 1, 14 )

points of intersection are (- 1, 14 ) and (3, 6 )