It one root of the equation \(\mathrm{x}^{2}+\mathrm{px}+12=0\) is 4 , while the equation \(\mathrm{x}^{2}+\mathrm{px}+\mathrm{q}=0\) has equal roots, then the value of \(\mathrm{q}\) is (a) \((49 / 4)\) (b) 12 (c) 3 (d) 4

Vieta's formulas are a set of equations that relate the coefficients of a polynomial to sums and products of its roots. Specifically, for a quadratic equation of the form \( ax^2 + bx + c = 0 \), the formulas state that the sum of the roots \( -\frac{b}{a} \) and the product of the roots \( \frac{c}{a} \).

For the quadratic equation \( x^2+px+12=0 \) given in the exercise, where the leading coefficient \( a = 1 \), Vieta's formulas tell us that if one root is 4, the other can be calculated as follows:

\begin{align*}\text{Sum of roots} = -p &= 4 + r \( r \) being the other root,\text{Product of roots} = 12 &= 4 \times r \end{align*}
By solving the product equation for \( r \), we can deduce that the second root is 3, and consequently find the value of \( p \) by using the sum equation. This brilliant insight comes from Vieta's elegant connection between the roots and coefficients of polynomials.

The discriminant of a quadratic equation provides critical information about the nature of its roots. For a general quadratic defined by \( ax^2 + bx + c = 0 \), the discriminant is \( D = b^2 - 4ac \) and can determine whether the quadratic has real or complex roots and whether those roots are distinct or repeated.

When \( D > 0 \), there are two distinct real roots. If \( D = 0 \), the equation has exactly one real root, meaning it is a repeated or double root. Lastly, if \( D < 0 \), the equation has two complex roots.

In the context of the given exercise, we explored the second equation, where we know the roots are equal. Therefore, the discriminant \( D \) must be zero. This condition allows us to establish a relationship between \( p \) and \( q \) and eventually solve for the unknown value of \( q \) using the derived formula for the discriminant.

The roots of a quadratic equation \( ax^2 + bx + c = 0 \) are the solutions to the equation and are calculated using the quadratic formula: \(\frac{-b \pm \sqrt{D}}{2a}\), where \( D \) is the discriminant. If the discriminant is zero (\( D = 0 \)), the quadratic formula shows us that there is only one root, or rather, a repeated root, because the term \( \pm \sqrt{D} \) becomes zero.

In the exercise we are considering, the second equation is said to have equal roots, meaning the discriminant is zero. This implies that both roots have the same value, further supporting our findings obtained through Vieta's formulas and the discriminant. Consequently, the single root's value is tightly integrated with the coefficients \( p \) and \( q \) of the quadratic, which influences our approach to finding the unknown \( q \) given the condition of equal roots.