To determine whether the roots of the quadratic equation $x^2 - 4abx + (a^2 + 2b^2)^2 = 0$ are imaginary, we need to analyze its discriminant.

A quadratic equation of the form $ax^2 + bx + c = 0$ has a discriminant given by:

$D = b^2 - 4ac$

If the discriminant $D < 0$, the roots are imaginary (complex numbers).

Step-by-Step Solution:

Given the equation:

$x^2 - 4abx + (a^2 + 2b^2)^2 = 0$

we identify the coefficients:

• $a = 1$ (coefficient of $x^2$)
• $b = -4ab$ (coefficient of $x$)
• $c = (a^2 + 2b^2)^2$ (constant term)

The discriminant $D$ is:

$D = (-4ab)^2 - 4 \cdot 1 \cdot (a^2 + 2b^2)^2$

Simplify each term:

$D = (16a^2b^2) - 4(a^2 + 2b^2)^2$

Now, expand the square in the second term:

$(a^2 + 2b^2)^2 = a^4 + 4a^2b^2 + 4b^4$

Substitute this back into the discriminant:

$D = 16a^2b^2 - 4(a^4 + 4a^2b^2 + 4b^4)$

Distribute the $4$ in the second term:

$D = 16a^2b^2 - (4a^4 + 16a^2b^2 + 16b^4)$

Combine like terms:

$D = 16a^2b^2 - 4a^4 - 16a^2b^2 - 16b^4$

$D = -4a^4 - 16b^4$

Factor out $-4$:

$D = -4(a^4 + 4b^4)$

Analysis of the Discriminant:

The discriminant $D = -4(a^4 + 4b^4)$. Since $a^4 \geq 0$ and $b^4 \geq 0$ for all real numbers $a$ and $b$, we have:

$a^4 + 4b^4 \geq 0$

Therefore, the expression $-4(a^4 + 4b^4)$ is always less than or equal to zero. Since $a^4 + 4b^4 > 0$ for any non-zero $a$ or $b$, we have:

$D < 0$

Conclusion:

The discriminant $D < 0$ shows that the roots of the equation $x^2 - 4abx + (a^2 + 2b^2)^2 = 0$ are imaginary.

Would you like more details or have any questions?

Related Questions:

1. How do you find the roots of a quadratic equation when the discriminant is negative?
2. What are the general properties of the roots of a quadratic equation?
3. How do imaginary roots affect the graph of a quadratic equation?
4. How can you identify if a quadratic equation has complex roots without calculating the discriminant?
5. What is the significance of the discriminant in quadratic equations?

Tip: Remember, the discriminant is a powerful tool to quickly determine the nature of the roots of any quadratic equation!