Determine The Number And Type Of Solutions For: 3 X ( X − 2 ) = X − 5 3x(x-2) = X-5 3 X ( X − 2 ) = X − 5 A. Two Rational Solutions B. Two Irrational Solutions C. One Rational (a Double Solution) D. Nonreal Complex Solutions

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Introduction

When solving a quadratic equation, it's essential to determine the number and type of solutions it has. This information can help us understand the behavior of the equation and make predictions about its graph. In this article, we'll explore the different types of solutions that a quadratic equation can have and how to determine them.

What are Rational and Irrational Solutions?

Before we dive into the specifics of the given equation, let's define what rational and irrational solutions are.

• Rational solutions are solutions that can be expressed as a fraction, i.e., a ratio of two integers. For example, 3/4 or -2/3 are rational solutions.
• Irrational solutions are solutions that cannot be expressed as a fraction. They are often expressed as decimals that go on forever without repeating. For example, √2 or π are irrational solutions.

What are Complex Solutions?

In addition to rational and irrational solutions, quadratic equations can also have complex solutions. Complex solutions are solutions that involve imaginary numbers, which are numbers that, when squared, give a negative result. For example, 3 + 4i or 2 - 3i are complex solutions.

The Given Equation

Now that we've covered the basics, let's take a look at the given equation:

$3x(x-2) = x-5$

Step 1: Expand the Left Side of the Equation

To begin solving the equation, we need to expand the left side:

$3x(x-2) = 3x^2 - 6x$

Step 2: Rewrite the Equation

Now that we've expanded the left side, we can rewrite the equation as:

$3x^2 - 6x = x-5$

Step 3: Move All Terms to One Side

To make it easier to solve the equation, let's move all the terms to one side:

$3x^2 - 6x - x + 5 = 0$

Step 4: Combine Like Terms

Now that we've moved all the terms to one side, let's combine like terms:

$3x^2 - 7x + 5 = 0$

Step 5: Determine the Number and Type of Solutions

To determine the number and type of solutions, we need to examine the discriminant of the quadratic equation. The discriminant is the expression under the square root in the quadratic formula:

$b^2 - 4ac$

In this case, a = 3, b = -7, and c = 5. Plugging these values into the discriminant, we get:

$(-7)^2 - 4(3)(5) = 49 - 60 = -11$

Since the discriminant is negative, we know that the equation has nonreal complex solutions.

Conclusion

In this article, we've explored the different types of solutions that a quadratic equation can have and how to determine them. We've also applied this knowledge to the given equation $3x(x-2) = x-5$ and determined that it has nonreal complex solutions.

Final Answer

The final answer is:

D. Nonreal complex solutions

Introduction

In our previous article, we explored the different types of solutions that a quadratic equation can have and how to determine them. In this article, we'll answer some frequently asked questions (FAQs) about determining the number and type of solutions for a quadratic equation.

Q: What is the discriminant, and why is it important?

A: The discriminant is the expression under the square root in the quadratic formula:

$b^2 - 4ac$

It's essential to determine the number and type of solutions because it helps us understand the behavior of the equation and make predictions about its graph.

Q: How do I determine the number and type of solutions for a quadratic equation?

A: To determine the number and type of solutions, you need to examine the discriminant of the quadratic equation. If the discriminant is:

• Positive, the equation has two rational solutions.
• Zero, the equation has one rational (a double solution).
• Negative, the equation has nonreal complex solutions.

Q: What are rational solutions, and how do I identify them?

A: Rational solutions are solutions that can be expressed as a fraction, i.e., a ratio of two integers. To identify rational solutions, you need to find the factors of the constant term (c) and the coefficient of the linear term (b). If the factors are integers, then the equation has rational solutions.

Q: What are irrational solutions, and how do I identify them?

A: Irrational solutions are solutions that cannot be expressed as a fraction. To identify irrational solutions, you need to examine the discriminant. If the discriminant is negative, then the equation has nonreal complex solutions, which are often irrational.

Q: What are complex solutions, and how do I identify them?

A: Complex solutions are solutions that involve imaginary numbers, which are numbers that, when squared, give a negative result. To identify complex solutions, you need to examine the discriminant. If the discriminant is negative, then the equation has nonreal complex solutions.

Q: How do I apply the quadratic formula to find the solutions of a quadratic equation?

A: The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

To apply the quadratic formula, you need to plug in the values of a, b, and c into the formula and simplify.

Q: What are some common mistakes to avoid when determining the number and type of solutions for a quadratic equation?

A: Some common mistakes to avoid include:

• Not examining the discriminant carefully.
• Not identifying the correct type of solution (rational, irrational, or complex).
• Not applying the quadratic formula correctly.

Conclusion

In this article, we've answered some frequently asked questions (FAQs) about determining the number and type of solutions for a quadratic equation. We've also provided some tips and common mistakes to avoid when working with quadratic equations.

Final Answer

The final answer is:

There is no final numerical answer to this article. However, the answers to the FAQs provided above should help you understand how to determine the number and type of solutions for a quadratic equation.