Consider The Quadratic Equation − 3 X 2 + 5 X − 2 = 0 -3x^2 + 5x - 2 = 0 − 3 X 2 + 5 X − 2 = 0 . Which Of The Following Statements Are True? Select All That Apply.A. The Discriminant Is A Positive, Perfect Square.B. The Discriminant Is Zero.C. There Are Two Rational Roots.D. There Is

Introduction

The quadratic equation is a fundamental concept in algebra, and it has numerous applications in various fields, including physics, engineering, and economics. In this article, we will analyze the quadratic equation $-3x^2 + 5x - 2 = 0$ and determine which of the given statements are true.

Understanding the Quadratic Equation

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

In the given equation $-3x^2 + 5x - 2 = 0$, we have $a = -3$, $b = 5$, and $c = -2$. To solve this equation, we can use various methods, including factoring, completing the square, and the quadratic formula.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that the solutions to the equation $ax^2 + bx + c = 0$ are given by:

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

In our case, we have $a = -3$, $b = 5$, and $c = -2$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-5 \pm \sqrt{5^2 - 4(-3)(-2)}}{2(-3)}$$

Simplifying the expression under the square root, we get:

$$x = \frac{-5 \pm \sqrt{25 - 24}}{-6}$$

$$x = \frac{-5 \pm \sqrt{1}}{-6}$$

$$x = \frac{-5 \pm 1}{-6}$$

Therefore, the solutions to the equation are:

$$x = \frac{-5 + 1}{-6} = \frac{-4}{-6} = \frac{2}{3}$$

$$x = \frac{-5 - 1}{-6} = \frac{-6}{-6} = 1$$

Analyzing the Statements

Now that we have solved the equation, let's analyze the given statements:

A. The discriminant is a positive, perfect square.

The discriminant is the expression under the square root in the quadratic formula, which is $b^2 - 4ac$. In our case, we have $b^2 - 4ac = 5^2 - 4(-3)(-2) = 25 - 24 = 1$. Since the discriminant is a positive, perfect square, this statement is TRUE.

B. The discriminant is zero.

As we have seen earlier, the discriminant is $1$, which is not zero. Therefore, this statement is FALSE.

C. There are two rational roots.

We have found two solutions to the equation, which are $x = \frac{2}{3}$ and $x = 1$. Since both solutions are rational numbers, this statement is TRUE.

D. There is only one rational root.

Since we have found two rational roots, this statement is FALSE.

Conclusion

In conclusion, the statements that are true are:

• The discriminant is a positive, perfect square.
• There are two rational roots.

The statements that are false are:

• The discriminant is zero.
• There is only one rational root.

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Introduction

In our previous article, we analyzed the quadratic equation $-3x^2 + 5x - 2 = 0$ and determined which of the given statements are true. In this article, we will provide a comprehensive Q&A guide to help you better understand quadratic equations and their solutions.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including:

• Factoring: This method involves expressing the quadratic equation as a product of two binomials.
• Completing the square: This method involves rewriting the quadratic equation in a form that allows us to easily find the solutions.
• The quadratic formula: This method involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula, which is $b^2 - 4ac$. The discriminant determines the nature of the solutions to the quadratic equation.

Q: What do the values of the discriminant mean?

A: The values of the discriminant can be:

• Positive: This means the quadratic equation has two distinct real solutions.
• Zero: This means the quadratic equation has one repeated real solution.
• Negative: This means the quadratic equation has no real solutions.

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

A: To find the solutions to a quadratic equation, you can use the quadratic formula:

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

Q: What is the difference between a rational root and an irrational root?

A: A rational root is a root that can be expressed as a fraction of two integers, while an irrational root is a root that cannot be expressed as a fraction of two integers.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions.

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions if the discriminant is negative.

Q: How do I determine the nature of the solutions to a quadratic equation?

A: To determine the nature of the solutions to a quadratic equation, you can use the discriminant. If the discriminant is positive, the quadratic equation has two distinct real solutions. If the discriminant is zero, the quadratic equation has one repeated real solution. If the discriminant is negative, the quadratic equation has no real solutions.

Conclusion

In conclusion, we hope this Q&A guide has provided a comprehensive understanding of quadratic equations and their solutions. Whether you are a student or a professional, understanding quadratic equations is essential for solving a wide range of mathematical problems.