What Is The Calculated Discriminant Of $3x^2 + X - 2 = 0$?

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Introduction

In mathematics, the discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is used to determine the nature of the roots of the equation, whether they are real or complex. The discriminant is calculated using the formula $D = b^2 - 4ac$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation. In this article, we will calculate the discriminant of the quadratic equation $3x^2 + x - 2 = 0$.

Understanding the Quadratic Equation

The given quadratic equation is $3x^2 + x - 2 = 0$. To calculate the discriminant, we need to identify the coefficients $a$, $b$, and $c$. In this equation, $a = 3$, $b = 1$, and $c = -2$.

Calculating the Discriminant

Now that we have identified the coefficients, we can calculate the discriminant using the formula $D = b^2 - 4ac$. Substituting the values of $a$, $b$, and $c$, we get:

D=(1)2−4(3)(−2)D = (1)^2 - 4(3)(-2)

D=1+24D = 1 + 24

D=25D = 25

Interpreting the Result

The calculated discriminant is $D = 25$. This value tells us that the quadratic equation $3x^2 + x - 2 = 0$ has two distinct real roots. The nature of the roots can be determined by the sign of the discriminant. If the discriminant is positive, the roots are real and distinct. If the discriminant is zero, the roots are real and equal. If the discriminant is negative, the roots are complex.

Conclusion

In this article, we calculated the discriminant of the quadratic equation $3x^2 + x - 2 = 0$. The calculated discriminant is $D = 25$, which indicates that the equation has two distinct real roots. This result can be used to determine the nature of the roots and to solve the equation.

Calculating the Roots

Now that we have calculated the discriminant, we can use it to calculate the roots of the equation. The roots of a quadratic equation can be calculated using the quadratic formula:

x=−b±D2ax = \frac{-b \pm \sqrt{D}}{2a}

Substituting the values of $a$, $b$, and $D$, we get:

x=−1±252(3)x = \frac{-1 \pm \sqrt{25}}{2(3)}

x=−1±56x = \frac{-1 \pm 5}{6}

This gives us two possible values for $x$:

x=−1+56=46=23x = \frac{-1 + 5}{6} = \frac{4}{6} = \frac{2}{3}

x=−1−56=−66=−1x = \frac{-1 - 5}{6} = \frac{-6}{6} = -1

Conclusion

In this article, we calculated the discriminant of the quadratic equation $3x^2 + x - 2 = 0$ and used it to calculate the roots of the equation. The calculated discriminant is $D = 25$, which indicates that the equation has two distinct real roots. The roots of the equation are $x = \frac{2}{3}$ and $x = -1$.

Importance of the Discriminant

The discriminant is an important concept in mathematics, particularly in algebra. It is used to determine the nature of the roots of a quadratic equation, whether they are real or complex. The discriminant is also used in various applications, such as physics, engineering, and computer science.

Real-World Applications

The discriminant has several real-world applications. In physics, the discriminant is used to determine the energy levels of a system. In engineering, the discriminant is used to design and optimize systems. In computer science, the discriminant is used in machine learning algorithms.

Conclusion

In conclusion, the discriminant is an important concept in mathematics that is used to determine the nature of the roots of a quadratic equation. The calculated discriminant of the quadratic equation $3x^2 + x - 2 = 0$ is $D = 25$, which indicates that the equation has two distinct real roots. The roots of the equation are $x = \frac{2}{3}$ and $x = -1$. The discriminant has several real-world applications, including physics, engineering, and computer science.

Final Thoughts

The discriminant is a powerful tool in mathematics that can be used to solve quadratic equations and determine the nature of their roots. It is an important concept that has several real-world applications. In this article, we calculated the discriminant of the quadratic equation $3x^2 + x - 2 = 0$ and used it to calculate the roots of the equation. We hope that this article has provided a clear understanding of the discriminant and its importance in mathematics.

Introduction

In the previous article, we discussed the concept of the discriminant and calculated its value for the quadratic equation $3x^2 + x - 2 = 0$. In this article, we will answer some frequently asked questions (FAQs) about the discriminant.

Q: What is the discriminant of a quadratic equation?

A: The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is used to determine the nature of the roots of the equation, whether they are real or complex.

Q: How is the discriminant calculated?

A: The discriminant is calculated using the formula $D = b^2 - 4ac$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: What does the discriminant tell us about the roots of a quadratic equation?

A: The discriminant tells us about the nature of the roots of a quadratic equation. If the discriminant is positive, the roots are real and distinct. If the discriminant is zero, the roots are real and equal. If the discriminant is negative, the roots are complex.

Q: Can the discriminant be negative?

A: Yes, the discriminant can be negative. This means that the roots of the quadratic equation are complex.

Q: Can the discriminant be zero?

A: Yes, the discriminant can be zero. This means that the roots of the quadratic equation are real and equal.

Q: Can the discriminant be positive?

A: Yes, the discriminant can be positive. This means that the roots of the quadratic equation are real and distinct.

Q: How is the discriminant used in real-world applications?

A: The discriminant is used in various real-world applications, such as physics, engineering, and computer science. In physics, the discriminant is used to determine the energy levels of a system. In engineering, the discriminant is used to design and optimize systems. In computer science, the discriminant is used in machine learning algorithms.

Q: Can the discriminant be used to solve quadratic equations?

A: Yes, the discriminant can be used to solve quadratic equations. The roots of a quadratic equation can be calculated using the quadratic formula, which involves the discriminant.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to calculate the roots of a quadratic equation. It is given by:

x=−b±D2ax = \frac{-b \pm \sqrt{D}}{2a}

Q: How is the quadratic formula used to calculate the roots of a quadratic equation?

A: The quadratic formula is used to calculate the roots of a quadratic equation by substituting the values of $a$, $b$, and $D$ into the formula.

Q: Can the discriminant be used to determine the nature of the roots of a quadratic equation?

A: Yes, the discriminant can be used to determine the nature of the roots of a quadratic equation. The discriminant tells us whether the roots are real or complex.

Q: Can the discriminant be used to calculate the roots of a quadratic equation?

A: Yes, the discriminant can be used to calculate the roots of a quadratic equation. The roots of a quadratic equation can be calculated using the quadratic formula, which involves the discriminant.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about the discriminant. We hope that this article has provided a clear understanding of the discriminant and its importance in mathematics.

Final Thoughts

The discriminant is a powerful tool in mathematics that can be used to determine the nature of the roots of a quadratic equation. It is an important concept that has several real-world applications. In this article, we answered some frequently asked questions (FAQs) about the discriminant and provided a clear understanding of its importance in mathematics.

Additional Resources

For more information about the discriminant, we recommend the following resources:

  • [1] Khan Academy: Quadratic Equations and the Discriminant
  • [2] Math Is Fun: Quadratic Formula and the Discriminant
  • [3] Wolfram MathWorld: Discriminant

We hope that this article has provided a clear understanding of the discriminant and its importance in mathematics.