Question

Work out the coordinates of the points of intersection of the graphs

y= 4x^2+8x+3 and y=x+5
Answer (_____,_____) and (______,______)

Answer 1

(- 2, 3 ) and (
`(1)/(4)`,
`(21)/(4)` )

Explanation:

Given the 2 equations

y = 4x² + 8x + 3 → (1)

y = x + 5 → (2)

Substitute y = 4x² + 8x + 3 into (2)

4x² + 8x + 3 = x + 5 ( subtract x + 5 from both sides )

4x² + 7x - 2 = 0 ← in standard form

(x + 2)(4x - 1) = 0 ← in factored form

Equate each factor to zero and solve for x

x + 2 = 0 ⇒ x = - 2

4x - 1 = 0 ⇒ 4x = 1 ⇒ x =
`(1)/(4)`

Substitute these values into (2) for corresponding values of y

x = - 2 : y = - 2 + 5 = 3 ⇒ (- 2, 3 )

x =
`(1)/(4)` : y =
`(1)/(4)` + 5 =
`(21)/(4)` ⇒ (
`(1)/(4)`,
`(21)/(4)` )

Answer 2

Explanation:

The point of intersection occurs when the two graphs have equal values of x and y at the same time. There is only one solution, because two straight lines can only intersect once.

`y =4x^2 + 8x + 3 , y = x + 5\\\\4x^2 + 8x + 3 = x +5\\\\4x^2 + 8x + 3 -x -5 = 0\\\\4x^2 +7x -2 =0\\\\4x^2 + 8x -x - 2 = 0\\\\4x(x + 2) -1(x+2) = 0\\\\(4x - 1)(x+2) =0\\\\4x -1 = 0 , x+2 =0\\\\x = (1)/(4), -2`

`x_1 = (1)/(4), x_2 =-2`

`y_1 = 4x^2 + 8x + 3 \\\\x_1 = (1)/(4)\\\\y_1 = 4((1)/(4))^2 + 8((1)/(4)) + 3 = (1)/(4) + 2 + 3 = (1)/(4) +5 = (21)/(4)\\\\y_2 = x+ 5\\x_2 = -2\\y_2 = -2 + 5 =3`

Answer :

`(x_1, y_1) = ((1)/(4) , (21)/(4))\\\\(x_2,y_2) = (-2, 3)`