Which Of The Statements About The Following Quadratic Equation Is True?${ 6x^2 - 8 = 4x^2 + 7x }$A. The Discriminant Is Greater Than Zero, So There Are Two Real Roots. B. The Discriminant Is Less Than Zero, So There Are Two Complex Roots.

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Understanding Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. 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.

The Given Quadratic Equation

The given quadratic equation is 6x^2 - 8 = 4x^2 + 7x. To analyze this equation, we need to rewrite it in the standard form of a quadratic equation, which is ax^2 + bx + c = 0. By rearranging the terms, we get:

6x^2 - 4x^2 - 7x - 8 = 0

Combine like terms:

2x^2 - 7x - 8 = 0

Analyzing the Quadratic Equation

Now that we have the quadratic equation in the standard form, we can analyze it further. The coefficients of the quadratic equation are a = 2, b = -7, and c = -8. To determine the nature of the roots, we need to calculate the discriminant, which is given by the formula b^2 - 4ac.

Calculating the Discriminant

Substitute the values of a, b, and c into the formula:

b^2 - 4ac = (-7)^2 - 4(2)(-8) = 49 + 64 = 113

Interpreting the Discriminant

The discriminant is 113, which is greater than zero. This means that the quadratic equation has two real roots. However, we need to determine which of the given statements is true.

Evaluating Statement A

Statement A claims that the discriminant is greater than zero, so there are two real roots. As we calculated earlier, the discriminant is indeed greater than zero, which supports this statement.

Evaluating Statement B

Statement B claims that the discriminant is less than zero, so there are two complex roots. However, we calculated the discriminant to be 113, which is greater than zero, not less than zero. Therefore, this statement is false.

Conclusion

Based on our analysis, we can conclude that statement A is true. The discriminant is indeed greater than zero, which means that the quadratic equation has two real roots.

Importance of Quadratic Equations

Quadratic equations are essential in various fields, including physics, engineering, and economics. They are used to model real-world problems, such as the motion of objects, the growth of populations, and the behavior of electrical circuits. Understanding quadratic equations is crucial for solving these problems and making informed decisions.

Real-World Applications of Quadratic Equations

Quadratic equations have numerous real-world applications, including:

  • Projectile Motion: Quadratic equations are used to model the motion of projectiles, such as the trajectory of a thrown ball or the path of a rocket.
  • Electrical Circuits: Quadratic equations are used to analyze the behavior of electrical circuits, including the voltage and current in a circuit.
  • Population Growth: Quadratic equations are used to model the growth of populations, including the growth of bacteria or the population of a city.
  • Optimization: Quadratic equations are used to optimize functions, including the minimization or maximization of a function.

Conclusion

In conclusion, the given quadratic equation has two real roots, and statement A is true. Quadratic equations are essential in various fields, and understanding them is crucial for solving real-world problems.

Frequently Asked Questions

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. In this article, we will answer some of the most frequently asked questions about quadratic equations.

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: If the quadratic equation can be factored into the product of two binomials, we can solve it by setting each factor equal to zero.
  • Quadratic Formula: The quadratic formula is a general method for solving quadratic equations. It is given by the formula x = (-b ± √(b^2 - 4ac)) / 2a.
  • Graphing: We can also solve a quadratic equation by graphing the related function and finding the x-intercepts.

Q: What is the discriminant, and how do I calculate it?

A: The discriminant is a value that can be calculated from the coefficients of a quadratic equation. It is given by the formula b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation. The discriminant can be used to determine the nature of the roots of the quadratic equation.

Q: What do the values of the discriminant mean?

A: The value of the discriminant can be used to determine the nature of the roots of the quadratic equation. If the discriminant is:

  • Positive: The quadratic equation has two real roots.
  • Zero: The quadratic equation has one real root.
  • Negative: The quadratic equation has two complex roots.

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

A: To determine the nature of the roots of a quadratic equation, we need to calculate the discriminant. If the discriminant is positive, the quadratic equation has two real roots. If the discriminant is zero, the quadratic equation has one real root. If the discriminant is negative, the quadratic equation has two complex roots.

Q: What are the applications of quadratic equations?

A: Quadratic equations have numerous applications in various fields, including:

  • Physics: Quadratic equations are used to model the motion of objects, including the trajectory of a thrown ball or the path of a rocket.
  • Engineering: Quadratic equations are used to analyze the behavior of electrical circuits, including the voltage and current in a circuit.
  • Economics: Quadratic equations are used to model the growth of populations, including the growth of bacteria or the population of a city.
  • Computer Science: Quadratic equations are used in computer graphics, including the rendering of 3D models.

Q: How do I use quadratic equations in real-world problems?

A: To use quadratic equations in real-world problems, we need to:

  • Model the problem: We need to identify the variables and the relationships between them.
  • Write the equation: We need to write the quadratic equation that models the problem.
  • Solve the equation: We need to solve the quadratic equation to find the values of the variables.
  • Interpret the results: We need to interpret the results of the solution to the quadratic equation.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. Understanding quadratic equations is crucial for solving real-world problems and making informed decisions.