What Is The Discriminant Of $9x^2 + 2 = 10x$?A. -356 B. -172 C. 28 D. 72

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Introduction to the Discriminant

The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation and provides information about the nature of the roots of the equation. In this article, we will explore the concept of the discriminant and how to calculate it for a given quadratic equation.

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. The general form of a quadratic equation is:

ax^2 + bx + c = 0

where a, b, and c are constants, and a is not equal to zero.

The Discriminant Formula

The discriminant of a quadratic equation is given by the formula:

Δ = b^2 - 4ac

where Δ is the discriminant, and a, b, and c are the coefficients of the quadratic equation.

Calculating the Discriminant

To calculate the discriminant, we need to substitute the values of a, b, and c into the formula. Let's consider the given quadratic equation:

9x^2 + 2 = 10x

We can rewrite this equation in the standard form as:

9x^2 - 10x + 2 = 0

Now, we can identify the values of a, b, and c:

a = 9 b = -10 c = 2

Substituting these values into the discriminant formula, we get:

Δ = (-10)^2 - 4(9)(2) Δ = 100 - 72 Δ = 28

Conclusion

The discriminant of the given quadratic equation is 28. This value provides information about the nature of the roots of the equation. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots.

Understanding the Options

Now that we have calculated the discriminant, let's compare it with the given options:

A. -356 B. -172 C. 28 D. 72

The correct answer is C. 28.

Real-World Applications of the Discriminant

The discriminant has many real-world applications, including:

  • Physics: The discriminant is used to determine the nature of the roots of a quadratic equation that represents the motion of an object under the influence of a force.
  • Engineering: The discriminant is used to determine the stability of a system represented by a quadratic equation.
  • Computer Science: The discriminant is used in algorithms for solving quadratic equations and in computer graphics for determining the nature of the roots of a quadratic equation.

Final Thoughts

In conclusion, the discriminant of a quadratic equation is a value that provides information about the nature of the roots of the equation. Calculating the discriminant involves substituting the values of the coefficients into the formula and simplifying the expression. The discriminant has many real-world applications and is an important concept in mathematics and science.

Frequently Asked Questions

  • What is the discriminant of a quadratic equation? The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation and provides information about the nature of the roots of the equation.
  • How do I calculate the discriminant? To calculate the discriminant, substitute the values of the coefficients into the formula and simplify the expression.
  • What is the significance of the discriminant? The discriminant provides information about the nature of the roots of the equation, including whether the equation has two distinct real roots, one real root, or no real roots.

References

  • "Algebra" by Michael Artin
  • "Calculus" by Michael Spivak
  • "Linear Algebra and Its Applications" by Gilbert Strang

Note: The references provided are for general information and are not specific to the topic of the discriminant.

Introduction

The discriminant is a fundamental concept in mathematics, particularly in algebra and geometry. It is a value that can be calculated from the coefficients of a quadratic equation and provides information about the nature of the roots of the equation. In this article, we will answer some of the most frequently asked questions about the discriminant.

Q&A

Q1: What is the discriminant of a quadratic equation?

A1: The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation and provides information about the nature of the roots of the equation.

Q2: How do I calculate the discriminant?

A2: To calculate the discriminant, substitute the values of the coefficients into the formula and simplify the expression. The formula for the discriminant is:

Δ = b^2 - 4ac

where Δ is the discriminant, and a, b, and c are the coefficients of the quadratic equation.

Q3: What is the significance of the discriminant?

A3: The discriminant provides information about the nature of the roots of the equation, including whether the equation has two distinct real roots, one real root, or no real roots.

Q4: What is the difference between the discriminant and the quadratic formula?

A4: The discriminant is a value that can be calculated from the coefficients of the quadratic equation, while the quadratic formula is a method for solving quadratic equations. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

The discriminant is used to determine the nature of the roots of the equation, while the quadratic formula is used to find the roots themselves.

Q5: Can the discriminant be negative?

A5: Yes, the discriminant can be negative. If the discriminant is negative, the equation has no real roots.

Q6: Can the discriminant be zero?

A6: Yes, the discriminant can be zero. If the discriminant is zero, the equation has one real root.

Q7: Can the discriminant be positive?

A7: Yes, the discriminant can be positive. If the discriminant is positive, the equation has two distinct real roots.

Q8: How is the discriminant used in real-world applications?

A8: The discriminant is used in many real-world applications, including physics, engineering, and computer science. It is used to determine the nature of the roots of a quadratic equation, which is essential in many fields.

Q9: Can the discriminant be used to solve quadratic equations?

A9: Yes, the discriminant can be used to solve quadratic equations. If the discriminant is positive, the equation has two distinct real roots, which can be found using the quadratic formula.

Q10: Can the discriminant be used to determine the stability of a system?

A10: Yes, the discriminant can be used to determine the stability of a system represented by a quadratic equation. If the discriminant is negative, the system is unstable.

Conclusion

The discriminant is a fundamental concept in mathematics, particularly in algebra and geometry. It is a value that can be calculated from the coefficients of a quadratic equation and provides information about the nature of the roots of the equation. In this article, we have answered some of the most frequently asked questions about the discriminant.

Final Thoughts

The discriminant is a powerful tool in mathematics and has many real-world applications. It is used to determine the nature of the roots of a quadratic equation, which is essential in many fields. We hope that this article has provided a comprehensive overview of the discriminant and its applications.

References

  • "Algebra" by Michael Artin
  • "Calculus" by Michael Spivak
  • "Linear Algebra and Its Applications" by Gilbert Strang

Note: The references provided are for general information and are not specific to the topic of the discriminant.