Hello I am trying to solve the equation:
tan^(2)x + 2tanx -3 = 0
the first part says tangent squared times x
the second part is 2 times tangent x
tan^2x + 2tanx -3 = 0
(tanx-3)(tanx+1) = 0
tanx = 3
tanx = -1
Now just find where tanx has those values
To solve the equation tan^2(x) + 2tan(x) - 3 = 0, we can use a substitution.
Let's substitute tan(x) with a new variable, let's say u.
So, u = tan(x).
Now, we can rewrite the equation in terms of u as follows:
u^2 + 2u - 3 = 0.
To solve this quadratic equation, we can factor it or use the quadratic formula.
Let's try factoring:
(u + 3)(u - 1) = 0.
Setting each factor to zero, we get the following equations:
u + 3 = 0 --> u = -3
u - 1 = 0 --> u = 1.
Since we substituted tan(x) with u, we can back-substitute to find the values of x.
When u = -3:
tan(x) = -3.
To find x, you can either use a calculator or the inverse tangent function (tan^-1).
Using the inverse tangent function, we find:
x = tan^-1(-3) + nπ, where n is an integer.
When u = 1:
tan(x) = 1.
Similarly, to find x, you can use a calculator or the inverse tangent function.
Using the inverse tangent function, we find:
x = tan^-1(1) + nπ, where n is an integer.
So, the solutions to the equation tan^2(x) + 2tan(x) - 3 = 0 are:
x = tan^-1(-3) + nπ, and
x = tan^-1(1) + nπ, where n is an integer.
To solve the given equation:
1. Begin by letting u = tan(x). This substitution will simplify the equation.
Our equation now becomes: u^2 + 2u - 3 = 0.
2. Factor the quadratic equation u^2 + 2u - 3 = 0.
The factors are (u + 3)(u - 1) = 0.
3. Set each factor to zero and solve for u:
u + 3 = 0 or u - 1 = 0.
4. Solve for u in each equation:
For u + 3 = 0, subtract 3 from both sides: u = -3.
For u - 1 = 0, add 1 to both sides: u = 1.
5. Recall that u = tan(x) and substitute back:
For u = -3, tan(x) = -3.
For u = 1, tan(x) = 1.
6. To find the solutions for x, take inverse tangent (arctan) of both sides:
For tan(x) = -3, x = arctan(-3).
For tan(x) = 1, x = arctan(1).
7. Use a calculator to find the arctan values:
x ≈ -1.25 + πn, where n is an integer (arctan(-3) in radians)
x ≈ 0.785 + πn, where n is an integer (arctan(1) in radians)
Therefore, the general solutions for x are:
x ≈ -1.25 + πn, where n is an integer (in radians)
x ≈ 0.785 + πn, where n is an integer (in radians)