Question

If one root of $x^{2} - 7 + 12 = 0$ is $4$, while the equation $x^{2} - 7x + q = 0$ has equal roots, then the value of $q$ is:

• A. $\frac{49}{4}$
• B. $\frac{4}{49}$
• C. $4$
• D. $\frac{1}{4}$

Hint:

When a quadratic has equal roots, set the discriminant to zero to find the relationship between coefficients. Always verify given root information by checking the sum and product of roots.

Answer 1

The Correct Option is A

First, the equation given is: $x^{2} - 7 + 12 = 0$. However, it seems like there is a typographical error — it should be $x^{2} - 7x + 12 = 0$ for the question to make sense.
We are told one root is $4$. Let's find the other root using the sum and product of roots formula.
From $x^{2} - 7x + 12 = 0$:
Sum of roots = $7$.
If one root is $4$, the other root = $7 - 4 = 3$.
Product of roots = $4 \times 3 = 12$ (matches the constant term).
Now, the second equation is: $x^{2} - 7x + q = 0$ and it has equal roots.
Condition for equal roots: Discriminant $\Delta = 0$.
So: $\Delta = (-7)^{2} - 4(1)(q) = 0$.
$49 - 4q = 0 \Rightarrow 4q = 49 \Rightarrow q = \frac{49}{4}$.
Thus, the value of $q$ is $\boxed{\frac{49}{4}}$.