What is the value of x in Arctan 3x + Arctan 2x = 45 deg?

The given equation is:

\[
\arctan(3x) + \arctan(2x) = 45^\circ
\]

We will use the following identity for the sum of two arctangents:

\[
\arctan(a) + \arctan(b) = \arctan\left(\frac{a + b}{1 - ab}\right)
\]

This identity holds when \(ab < 1\). Now, applying this to the equation:

\[
\arctan(3x) + \arctan(2x) = \arctan\left(\frac{3x + 2x}{1 - 3x \cdot 2x}\right)
\]

Simplifying the right-hand side:

\[
\arctan\left(\frac{5x}{1 - 6x^2}\right) = 45^\circ
\]

We know that \(\arctan(1) = 45^\circ\), so:

\[
\frac{5x}{1 - 6x^2} = 1
\]

Now, solve for \(x\):

\[
5x = 1 - 6x^2
\]

Rearranging the equation:

\[
6x^2 + 5x - 1 = 0
\]

This is a quadratic equation. We will solve it using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

For \(6x^2 + 5x - 1 = 0\), \(a = 6\), \(b = 5\), and \(c = -1\). Substituting these values into the quadratic formula:

\[
x = \frac{-5 \pm \sqrt{5^2 - 4(6)(-1)}}{2(6)}
\]
\[
x = \frac{-5 \pm \sqrt{25 + 24}}{12}
\]
\[
x = \frac{-5 \pm \sqrt{49}}{12}
\]
\[
x = \frac{-5 \pm 7}{12}
\]

So, we have two possible solutions:

\[
x = \frac{-5 + 7}{12} = \frac{2}{12} = \frac{1}{6}
\]
\[
x = \frac{-5 - 7}{12} = \frac{-12}{12} = -1
\]

Thus, the two possible solutions for \(x\) are:

\[
x = \frac{1}{6} \quad \text{or} \quad x = -1
\]

However, we must check which solution satisfies the condition that \(ab < 1\). For \(x = -1\):

\[
3(-1) \cdot 2(-1) = 6 > 1
\]

So, \(x = -1\) does not satisfy the condition. Hence, the only valid solution is:

\[
x = \frac{1}{6}
\]

Would you like further details or clarification on any part?

Here are 5 questions to deepen your understanding:
1. How does the sum identity for arctangents work in other cases?
2. What happens if we change the constants in \(\arctan(3x)\) and \(\arctan(2x)\)?
3. How can we verify that \(x = \frac{1}{6}\) is the correct solution?
4. Can arctangent equations have more than two solutions in other cases?
5. Why do we discard the solution \(x = -1\)?

**Tip**: Always check the conditions for trigonometric identities, especially for values that might be restricted by the domain.

Discover how to solve the equation Arctan(3x) + Arctan(2x) = 45° with step-by-step explanations. This problem covers important trigonometric identities and quadratic equations, ideal for students in Grades 10-12.

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