Show that the roots of the equation x²-4abx+(a²+2b²)²=0 are imaginary

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!

Show that the roots of the equation x²-4abx+(a²+2b²)²=0 are imaginary.

Learn how to show that the roots of the quadratic equation x² - 4abx + (a² + 2b²)² = 0 are imaginary by analyzing its discriminant. This step-by-step solution explores quadratic equations and the discriminant formula, suitable for Grades 10-12.

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