SOLUTION: Kuta Software Infinite Algebra 2: Understanding the ... - Free Printable

It looks like the image you uploaded contains a math problem involving quadratic equations and their roots, with instructions to determine whether the roots are real, rational, irrational, or complex based on the discriminant.

However, due to the blurriness of the image, I can't read the exact equations or details clearly. But based on the visible structure and common patterns in such problems, I’ll explain how to solve this type of problem step-by-step and provide guidance on determining the nature of the roots of a quadratic equation.

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🔍 General Approach: Nature of Roots of a Quadratic Equation

For a quadratic equation in the form:

$$
ax^2 + bx + c = 0
$$

The discriminant is:

$$
D = b^2 - 4ac
$$

The nature of the roots depends on the value of $ D $:

| Discriminant ($ D $) | Nature of Roots |
|------------------------|------------------|
| $ D > 0 $ and perfect square | Two distinct rational real roots |
| $ D > 0 $ but not a perfect square | Two distinct irrational real roots |
| $ D = 0 $ | One repeated real root (rational) |
| $ D < 0 $ | Two complex conjugate roots (imaginary) |

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✔ Step-by-Step Solution Process

Let’s suppose one of the equations in your image is:

$$
x^2 - 4x + 4 = 0
$$

Then:
- $ a = 1 $, $ b = -4 $, $ c = 4 $
- $ D = (-4)^2 - 4(1)(4) = 16 - 16 = 0 $

Since $ D = 0 $, the roots are real and equal (i.e., repeated), and rational.

So you would write: "1 real root (repeated)"

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Another example:

$$
x^2 - 5x + 6 = 0
$$

- $ D = (-5)^2 - 4(1)(6) = 25 - 24 = 1 > 0 $, and 1 is a perfect square → two distinct rational roots.

Answer: "2 rational"

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Now, consider:

$$
x^2 + 2x + 5 = 0
$$

- $ D = (2)^2 - 4(1)(5) = 4 - 20 = -16 < 0 $

→ Roots are complex (imaginary)

Answer: "2 imaginary"

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🧩 What About Irrational Roots?

Example:

$$
x^2 - 3x + 1 = 0
$$

- $ D = 9 - 4 = 5 > 0 $, but 5 is not a perfect square → roots are irrational

Answer: "2 irrational"

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💡 Final Tips for Your Problem

Look at each equation and compute the discriminant:

1. Identify $ a $, $ b $, $ c $
2. Compute $ D = b^2 - 4ac $
3. Use the table above to classify the roots

You may also see phrases like:
- "2 distinct real roots"
- "no real roots" (meaning complex)
- "repeated root"

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❓ Additional Questions from Image?

At the bottom, it seems there's a challenge:

> "Show that the roots are always either positive and integer, or complex."

But without clear equations, I can’t verify this claim.

If you're working with an equation like:

$$
x^2 + ax + 1 = 0
$$

And asked to analyze its roots, then:
- $ D = a^2 - 4 $
- For real roots: $ a^2 \geq 4 $ → $ |a| \geq 2 $
- If $ a $ is integer and $ |a| \geq 2 $, roots could be rational or irrational depending on $ a^2 - 4 $ being a perfect square.

But if $ |a| < 2 $, then $ D < 0 $ → complex roots.

So yes, roots are either complex or real, and if real, they may or may not be integers.

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📌 Summary

To solve your problem:
1. For each quadratic equation, calculate $ D = b^2 - 4ac $
2. Use the sign and nature of $ D $ to classify the roots
3. Fill in the boxes accordingly:
- "2 real, rational"
- "2 real, irrational"
- "1 real (repeated)"
- "2 imaginary"

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🛠️ Request

If you can rewrite or describe the equations from the image (or upload a clearer version), I’d be happy to solve them exactly and fill in all the blanks for you.

Until then, use the method above — it works for all standard quadratic root classification problems.