Finding the Relationship Between Coefficients and Roots

The problem provides a quadratic equation \(x^2 + px + q = 0\). The roots of this equation are given as \( \tan 19^\circ \) and \( \tan 26^\circ \). We need to find the correct relationship between the coefficients \(p\) and \(q\).

Using Vieta's Formulas

For a quadratic equation of the form \(ax^2 + bx + c = 0\), Vieta's formulas state the relationship between the roots (\(\alpha\) and \(\beta\)) and the coefficients:

Sum of roots: \( \alpha + \beta = -\frac{b}{a} \)

Product of roots: \( \alpha \cdot \beta = \frac{c}{a} \)

In our equation \(x^2 + px + q = 0\), we have \(a=1\), \(b=p\), and \(c=q\). The roots are \( \alpha = \tan 19^\circ \) and \( \beta = \tan 26^\circ \). Applying Vieta's formulas:

Sum of roots: \( \tan 19^\circ + \tan 26^\circ = -\frac{p}{1} = -p \)

Product of roots: \( \tan 19^\circ \cdot \tan 26^\circ = \frac{q}{1} = q \)

So, we have two relationships:

$$ \tan 19^\circ + \tan 26^\circ = -p \quad \text{(Equation 1)} $$

$$ \tan 19^\circ \cdot \tan 26^\circ = q \quad \text{(Equation 2)} $$

Using a Trigonometric Identity

Notice that the sum of the angles of the roots is \( 19^\circ + 26^\circ = 45^\circ \). This suggests using the tangent addition formula:

$$ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} $$

Let \(A = 19^\circ\) and \(B = 26^\circ\). Then \(A + B = 19^\circ + 26^\circ = 45^\circ\).

Applying the formula:

$$ \tan(19^\circ + 26^\circ) = \frac{\tan 19^\circ + \tan 26^\circ}{1 - \tan 19^\circ \tan 26^\circ} $$

We know that \( \tan 45^\circ = 1 \). So,

$$ 1 = \frac{\tan 19^\circ + \tan 26^\circ}{1 - \tan 19^\circ \tan 26^\circ} $$

Substituting from Vieta's Formulas

Now, substitute the expressions for the sum and product of roots from Equation 1 and Equation 2 into the equation above:

Substitute \( \tan 19^\circ + \tan 26^\circ = -p \)

Substitute \( \tan 19^\circ \cdot \tan 26^\circ = q \)

The equation becomes:

$$ 1 = \frac{-p}{1 - q} $$

Solving for the Relationship between p and q

Now, we solve this equation for \(p\) and \(q\):

Multiply both sides by \( (1 - q) \):

$$ 1 \cdot (1 - q) = -p $$

$$ 1 - q = -p $$

Rearrange the terms to find the relationship:

$$ p - q = -1 $$

or

$$ q - p = 1 $$

Comparing with Options

Let's compare the derived relationship \( q - p = 1 \) with the given options:

Option 1: \( q - p = 1 \)

Option 2: \( p - q = 1 \)

Option 3: \( p + q = 2 \)

Option 4: \( p + q = 3 \)

The relationship \( q - p = 1 \) matches Option 1.

Conclusion

Using Vieta's formulas to relate the sum and product of the roots to the coefficients \(p\) and \(q\), and then using the tangent addition formula for the sum of the angles \(19^\circ\) and \(26^\circ\), we found that \( q - p = 1 \).