Comparing with the Quadratic Formula

Oct 13, 2025 by leolavanderia.com.br 129 views

Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding their properties is crucial for solving various problems in algebra and beyond. In this article, we will delve into the world of quadratic equations and explore the nature of their roots. Specifically, we will examine the given equation $3x^2 - 8x + 5 = 5x^2$ and determine which statement about it is true.

The Quadratic Formula

Before we dive into the specifics of the given equation, let's recall the quadratic formula, which is used to find the roots of a quadratic equation in the form $ax^2 + bx + c = 0$ . The quadratic formula is given by:

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

The expression under the square root, $b^2 - 4ac$ , is known as the discriminant. The nature of the roots of a quadratic equation depends on the value of the discriminant.

The Nature of Roots

The nature of the roots of a quadratic equation can be determined by the value of the discriminant. If the discriminant is:

Positive ( $b^2 - 4ac > 0$ ), the equation has two distinct real roots.
Zero ( $b^2 - 4ac = 0$ ), the equation has one real root (or two equal real roots).
Negative ( $b^2 - 4ac < 0$ ), the equation has no real roots (or two complex roots).

The Given Equation

Now, let's examine the given equation $3x^2 - 8x + 5 = 5x^2$ . To determine the nature of its roots, we need to rewrite it in the standard form $ax^2 + bx + c = 0$ . By subtracting $5x^2$ from both sides, we get:

$-2x^2 - 8x + 5 = 0$

Comparing with the Quadratic Formula

Comparing the given equation with the quadratic formula, we have:

$a = -2$ , $b = -8$ , and $c = 5$

Now, let's calculate the discriminant:

$b^2 - 4ac = (-8)^2 - 4(-2)(5) = 64 + 40 = 104$

Since the discriminant is positive ( $104 > 0$ ), we can conclude that the given equation has two distinct real roots.

Conclusion

In conclusion, the statement about the given equation $3x^2 - 8x + 5 = 5x^2$ that is true is:

The discriminant is greater than 0, so there are two real roots.

This is because the discriminant of the given equation is positive, indicating that it has two distinct real roots.

Final Thoughts

Introduction

Quadratic equations are a fundamental concept in mathematics, and understanding their properties is crucial for solving various problems in algebra and beyond. In our previous article, we explored the nature of roots of a quadratic equation and applied this knowledge to the given equation $3x^2 - 8x + 5 = 5x^2$ . In this article, we will continue to delve into the world of quadratic equations and provide a comprehensive Q&A guide to help you better understand this concept.

Q&A Guide

Q1: What is a quadratic equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. It is typically written in the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants.

Q2: What is the quadratic formula?

The quadratic formula is a mathematical formula used to find the roots of a quadratic equation. It is given by:

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

Q3: What is the discriminant?

The discriminant is the expression under the square root in the quadratic formula, given by $b^2 - 4ac$ . It determines the nature of the roots of a quadratic equation.

Q4: What are the possible values of the discriminant?

The discriminant can be:

Positive ( $b^2 - 4ac > 0$ ), indicating two distinct real roots.
Zero ( $b^2 - 4ac = 0$ ), indicating one real root (or two equal real roots).
Negative ( $b^2 - 4ac < 0$ ), indicating no real roots (or two complex roots).

Q5: How do I determine the nature of the roots of a quadratic equation?

To determine the nature of the roots of a quadratic equation, you need to calculate the discriminant and compare it to zero. If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has one real root (or two equal real roots). If it is negative, the equation has no real roots (or two complex roots).

Q6: What is the difference between a quadratic equation and a linear equation?

A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. A quadratic equation has a squared variable (usually x), while a linear equation does not.

Q7: Can a quadratic equation have more than two roots?

No, a quadratic equation can have at most two roots. This is because the quadratic formula only provides two possible values for the roots.

Q8: How do I solve a quadratic equation?

To solve a quadratic equation, you can use the quadratic formula or factorization. The quadratic formula is given by:

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

Factorization involves expressing the quadratic equation as a product of two binomials.

Q9: What is the significance of the quadratic formula?

The quadratic formula is a powerful tool for solving quadratic equations. It provides a general solution for any quadratic equation, regardless of the values of the coefficients.

Q10: Can I use the quadratic formula to solve a cubic equation?

No, the quadratic formula is only applicable to quadratic equations. It cannot be used to solve cubic equations or higher-degree polynomial equations.

Conclusion

In this article, we provided a comprehensive Q&A guide to help you better understand quadratic equations. We covered topics such as the definition of a quadratic equation, the quadratic formula, the discriminant, and the nature of roots. We also answered common questions about quadratic equations, including how to determine the nature of the roots and how to solve a quadratic equation. We hope this guide has been helpful in your understanding of quadratic equations.