Question

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

Short answer

The value of \(q\) is \(\frac{49}{4}\).

Step-by-step solution

Use the given root of the first equation to find the value of p

We know that one root of the equation \(x^2 + px + 12 = 0\) is 4. Since the product of roots is equal to 12, we can say that the second root is equal to 3, because \(4 \times 3 = 12\). According to Vieta's formulas, the sum of roots is equal to -p. So, we have \((4 + 3) = -p\). Now, let's solve for p: \( p = -(4 + 3) \) \( p = -7 \) Now we know that the value of p is -7.

Apply the property of equal roots in the second equation

The second equation is given by \(x^2 + px + q = 0\), which could be written now as \(x^2 - 7x + q = 0\) since we found p = -7 in the previous step. Since this equation has equal roots, the discriminant must be equal to 0. The discriminant, D, of a quadratic equation \(ax^2 + bx + c\) is given by \(D = b^2 - 4ac\). In this case, a=1, b=-7, and c=q. Now let's substitute the values and find the discriminant: \(D = (-7)^2 - 4 \times 1 \times q\)

Solve for q

Now we know that \(D = 0\), let's substitute this back into the discriminant equation and solve for q: \(0 = (-7)^2 - 4 \times 1 \times q\) \(0 = 49 - 4q\) Now, solve for q: \(4q = 49\) \(q = \frac{49}{4}\)

Conclusion

The value of q is \(\frac{49}{4}\). Therefore, the correct answer is (a).

Key concepts

Vieta's formulas

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.

Discriminant of a quadratic equation

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.

Roots of a quadratic equation

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.