Question 29

Explanation

We recall the concept of quadratic equations. For a quadratic equation $ax^2 + bx + c = 0$ with roots $x_1$ and $x_2$, we have

Let the roots be $a$ and $b$. Since one root is three times the other, the roots are $a$ and $3a$ (because $b = 3a$).

Sum of roots

Product of roots

Substitute equation (1) into (2)

So $k = -4$ or $k = \frac{1}{2}$.

Now we check whether both values of $k$ satisfy the condition that all roots are greater than $2$.

For $k = -4$

Since $a = -3 < 2$, then $k = -4$ does not satisfy the condition.

For $k = \frac{1}{2}$

Since $a = \frac{3}{2} < 2$, then $k = \frac{1}{2}$ also does not satisfy the condition.

Both values of $k$ do not satisfy the condition because the root $a = k + 1$ has a value less than $2$. Therefore, no value satisfies.