Key Concepts and Formulas

• Root Property: If $r$ is a root of the quadratic equation $ax^2 + bx + c = 0$, then $ar^2 + br + c = 0$.
• Discriminant: For a quadratic equation $ax^2 + bx + c = 0$, the discriminant is $D = b^2 - 4ac$.
• Condition for Equal Roots: A quadratic equation has equal roots if and only if its discriminant is zero, i.e., $b^2 - 4ac = 0$.

Step-by-Step Solution

Step 1: Determine the value of 'p' using the first equation.

We are given that $x=4$ is a root of the equation $x^2 + px + 12 = 0$.

• Why: Substituting the known root into the equation allows us to solve for the unknown coefficient 'p'. This value of 'p' is crucial as it links the two given equations.

Substitute $x=4$ into the first equation: $(4)^2 + p(4) + 12 = 0$ $16 + 4p + 12 = 0$ Combine the constant terms: $28 + 4p = 0$ Isolate 'p': $4p = -28$ $p = \frac{-28}{4}$ $p = -7$

Step 2: Apply the condition for equal roots to the second equation.

The second equation is $x^2 + px + q = 0$. We know that $p = -7$, so the equation becomes: $x^2 - 7x + q = 0$ We are given that this equation has equal roots.

• Why: The condition for equal roots ($D=0$) provides a direct relationship between the coefficients of the quadratic equation. Applying this allows us to establish an equation involving 'q'.

For the equation $x^2 - 7x + q = 0$:

• The coefficient of $x^2$ is $a = 1$.
• The coefficient of $x$ is $b = -7$.
• The constant term is $c = q$.

Using the discriminant formula for equal roots ($b^2 - 4ac = 0$): $(-7)^2 - 4(1)(q) = 0$ Calculate $(-7)^2$: $49 - 4q = 0$

Step 3: Solve for 'q'.

We have the equation derived from the equal roots condition: $49 - 4q = 0$

• Why: This is a simple linear equation in 'q'. Solving it will give us the final answer.

Rearrange the equation to solve for 'q': $4q = 49$ $q = \frac{49}{4}$

Common Mistakes & Tips

• Sign Errors: Pay close attention to negative signs, especially when squaring negative numbers or substituting into formulas.
• Coefficient Identification: Correctly identify the coefficients $a$, $b$, and $c$ for each quadratic equation.
• Condition Confusion: Ensure that the root $x=4$ is used only for the first equation and the "equal roots" condition only for the second equation.

Summary

This problem demonstrates how knowing a root of one quadratic equation helps determine a common coefficient, which then aids in analyzing another related quadratic equation. By applying the properties of roots and the condition for equal roots (discriminant equal to zero), we successfully determined that the value of '$q$' is $\frac{49}{4}$.

Final Answer

The final answer is \boxed{\frac{49}{4}}, which corresponds to option (D).