How do you use the quadratic formula to solve 3tan^2x+3tanx-4=0 in the interval [0,2pi)?

1 Answer
Nghi N
Nov 16, 2017

37^@17; 119^@60; 217^@17; 299^@60

Explanation:

f(x) = 3tan^2 x + 3tan x - 4 = 0.
Solve this quadratic equation for tan x by using the improved formula (Socratic, Google Search):
D = d^2 = b^2 - 4ac = 9 + 48 = 57 --> d = +- sqrt57 = +- 7.55
There are 2 real roots:
tan x = -b/(2a) +- d/(2a) = - 3/6 +- 7.55/6 = (-3 +- 7.55)/6
tan x = - 10.55/6 = - 1.76
tan x = 4.55/6 = 0.76
a. tan x = - 1.76
Calculator and unit circle give:
x = - 60^@40 and x = -60.40 + 180 = 119^@60
The arc (-60^@40) is co-terminal to arc
(360 - 60.40 = 299^@60)
b. tan x = 0.76
x = 37^@17 and x = 37.17 + 180 = 217^@17
Answers for (0, 360):
#119^@60; 299^@60; 37^@17; 217^@17

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