Question

Let $\alpha$ and $\beta$ be the two distinct roots of the equation $2x^2 - 8x + k = 0$. If $(\alpha + \beta)$ and $\alpha\beta$ are the distinct roots of the equation $x^2 + ax + a = 0$, what is the value of $5(k - a)$?

• A. 9
• B. 8
• C. 15
• D. 12

Answer 1

Answer: (B)

8

Answer 2

Let $\alpha$ and $\beta$ be the two distinct roots of the equation $2x^2 - 8x + k = 0$.
According to Vieta's formulas, the sum of the roots is $\alpha + \beta = -\frac{-8}{2} = \frac{8}{2} = 4$.
The product of the roots is $\alpha\beta = \frac{k}{2}$.

The problem states that $(\alpha + \beta)$ and $\alpha\beta$ are the distinct roots of the equation $x^2 + ax + a = 0$.
Let the roots of the second equation be $r_1 = \alpha + \beta$ and $r_2 = \alpha\beta$.
So, $r_1 = 4$ and $r_2 = \frac{k}{2}$.
Since the roots are distinct, $r_1 \neq r_2$, which means $4 \neq \frac{k}{2}$, or $k \neq 8$.

According to Vieta's formulas for the second equation $x^2 + ax + a = 0$:
The sum of the roots is $r_1 + r_2 = -a$.
Substituting the values of $r_1$ and $r_2$:
$4 + \frac{k}{2} = -a$ (Equation 1)

The product of the roots is $r_1 r_2 = a$.
Substituting the values of $r_1$ and $r_2$:
$4 \left(\frac{k}{2}\right) = a$
$2k = a$ (Equation 2)

Now we have a system of two linear equations with two variables $k$ and $a$. Substitute the expression for $a$ from Equation 2 into Equation 1:
$4 + \frac{k}{2} = -(2k)$
$4 + \frac{k}{2} = -2k$
To eliminate the fraction, multiply the entire equation by 2:
$2 \times 4 + 2 \times \frac{k}{2} = 2 \times (-2k)$
$8 + k = -4k$
Add $4k$ to both sides:
$8 + k + 4k = -4k + 4k$
$8 + 5k = 0$
Subtract 8 from both sides:
$5k = -8$
Divide by 5:
$k = -\frac{8}{5}$.

Now substitute the value of $k$ back into Equation 2 to find $a$:
$a = 2k = 2 \left(-\frac{8}{5}\right) = -\frac{16}{5}$.

We are asked to find the value of $5(k - a)$.
First, calculate $k - a$:
$k - a = -\frac{8}{5} - \left(-\frac{16}{5}\right)$
$k - a = -\frac{8}{5} + \frac{16}{5}$
$k - a = \frac{-8 + 16}{5}$
$k - a = \frac{8}{5}$.

Now, calculate $5(k - a)$:
$5(k - a) = 5 \times \left(\frac{8}{5}\right)$
$5(k - a) = 8$.

Therefore, the value of $5(k - a)$ is 8.