Finding tan(α + β) from Quadratic Equation Roots

The question provides a quadratic equation \(x^2 + bx + c = 0\) where tan α and tan β are the roots. We are asked to find the value of tan (α + β).

Let the roots of the quadratic equation \(ax^2 + bx + c = 0\) be \(r_1\) and \(r_2\). According to Vieta's formulas:

• Sum of roots: \(r_1 + r_2 = -\frac{b}{a}\)
• Product of roots: \(r_1 \cdot r_2 = \frac{c}{a}\)

In the given equation, \(x^2 + bx + c = 0\), the coefficient of \(x^2\) is \(a = 1\), the coefficient of \(x\) is \(b\), and the constant term is \(c\).

The roots are given as tan α and tan β. So, we have:

• Sum of roots: tan α + tan β = \(-\frac{b}{1} = -b\)
• Product of roots: tan α \(\cdot\) tan β = \(-\frac{c}{1} = c\)

Now, we need to find tan (α + β). We use the trigonometric identity for the tangent of a sum of angles:

\[ \tan (\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \cdot \tan \beta} \]

Substitute the values we found for the sum and product of the roots into this identity:

\[ \tan (\alpha + \beta) = \frac{-b}{1 - c} \]

We can rewrite this expression to match the format of the given options. Note that \(\frac{-b}{1 - c} = \frac{-b}{-(c - 1)} = \frac{b}{c - 1}\).

Using negative exponents, \(\frac{1}{c - 1}\) can be written as \((c - 1)^{-1}\). Therefore,

\[ \tan (\alpha + \beta) = b(c - 1)^{-1} \]

Comparing this result with the given options, we see that it matches option 4.

Revision Table: Quadratic Roots and Trigonometric Identities

Additional Information: Properties of Quadratic Equations and Tangent Identity

A quadratic equation of the form \(ax^2 + bx + c = 0\) (where \(a \ne 0\)) has at most two roots. These roots can be real or complex. Vieta's formulas provide a direct link between the coefficients of a polynomial equation and the sums and products of its roots. For a quadratic equation, these formulas are particularly simple and useful, as demonstrated in this problem.

The tangent addition formula, \(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \cdot \tan B}\), is a fundamental identity in trigonometry derived from the sine and cosine addition formulas. It is frequently used in problems involving angles that are sums or differences of other angles, especially when the tangents of the individual angles are known.

In this problem, the roots of the quadratic equation happened to be trigonometric values (tan α and tan β), which allowed us to connect the algebraic properties of quadratic equations (sum and product of roots) with trigonometric identities to find the required value.