what the answer to the question below

Question

At which of the following points do the two equations f(x) = 3x² + 5 and g(x) = 4x + 4 intersect?

Option A: (1, 8)

Option B: (8, 1)

Option C: (0, 5)

Option D: (0, 4)

Answer 1

A) (1,8) is the point of intersection of the two lines

a parabola and a straight line, which has two intersection points

3x^2 +5 = y

and

4x + 4 = y

3x^2 +5 = 4x +4

3x^2 -4x +1 = 0

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

x = 1/3 or 1

y = 16/3 or 8

two intersection points (1,8) and (1/3, 16/3)

an easier way to solve this problem is just

plug in the points into the equations and see which work

3x^2 +5= 4x+4

3(1)^2 +5 = 4(1) +4

3+5 = 4+4

8 = 8

immediatly A) is the correct Answer

try the others, none work

try C) (0,5) 3(0)^2 + 5 =5 doesn't = 4(0)+4=4

try B) (8,1) 3(8)^2 +5 =53 doesn't = 4(8)+4=36

try D) (0,4) 3(0)^2+5= 5 doesn't = 4(4) +4 = 20

Answer 2

Given options, I would start by plugging in those options to find the intersection.

A) 3(1)² + 5 = 8 and 4(1) + 4 = 8; this is an intersection

B) 3(8)² + 5 ≠ 1; this is not on the graph f(x)

C) 4(0) + 4 ≠ 5; this is not on the graph g(x)

D) 3(0)² + 5 ≠ 4; this is not on the graph f(x)

This means that option (A) is the correct answer.

Now, if we didn't have those options, you could instead set f(x) = g(x) and solve.

3x² + 5 = 4x + 4

3x² - 4x + 1 = 0

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

x=1 or x=¹/₃

Then find your points by plugging into one of the functions (I'll use g(x) as it's simpler).

g(1) = 4(1) + 4 = 8; (1,8) is a point of intersection

g(¹/₃) = 4(¹/₃) + 4 = ¹⁶/₃; (¹/₃,¹⁶/₃) is a point of intersection