If α, β are roots of x2 – 3ax + a2 = 0, find the value(s) of a if α2 + β2 = 74 .

1. Break Down the Problem

To solve the problem, we need to find the values of $a$ such that the roots $\alpha$ and $\beta$ of the quadratic equation $x^2 - 3ax + a^2 = 0$ satisfy the condition $\alpha^2 + \beta^2 = 74$.

2. Relevant Concepts

Using Vieta's formulas, we know:

• The sum of the roots $\alpha + \beta = 3a$
• The product of the roots $\alpha \beta = a^2$

We can express $\alpha^2 + \beta^2$ in terms of the sum and product of the roots: $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$

3. Analysis and Detail

1. Substitute Vieta’s formulas into the equation: $\alpha^2 + \beta^2 = (3a)^2 - 2(a^2)$ $\alpha^2 + \beta^2 = 9a^2 - 2a^2 = 7a^2$

2. Set this equal to the given condition: $7a^2 = 74$

3. Solve for $a^2$: $a^2 = \frac{74}{7}$

4. Since we require $a$, take the square root: $a = \pm \sqrt{\frac{74}{7}} = \pm \frac{\sqrt{74}}{\sqrt{7}} = \pm \frac{\sqrt{74 \cdot 7}}{7} = \pm \frac{\sqrt{518}}{7}$

4. Verify and Summarize

The roots of the quadratic give us the expression for $\alpha^2 + \beta^2$. The calculations yield $a$ as $\pm \frac{\sqrt{518}}{7}$.

Final Answer

The values of $a$ are: $a = \pm \frac{\sqrt{518}}{7}$