Compute The Value Of The Discriminant And Give The Number Of Real Solutions Of The Quadratic Equation.Equation: { -4x^2 + 7x - 1 = 0$}$Discriminant: { \square$}$Number Of Real Solutions: { \square$}$

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. 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 this article, we will focus on solving the quadratic equation ${-4x^2 + 7x - 1 = 0}$ and compute the value of the discriminant and the number of real solutions.

The Quadratic Formula

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

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

This formula is derived from the fact that the quadratic equation can be factored as ${(x - r_1)(x - r_2) = 0}$, where ${r_1}$ and ${r_2}$ are the roots of the equation.

Computing the Discriminant

The discriminant is the expression under the square root in the quadratic formula, which is ${b^2 - 4ac}$. In our equation ${-4x^2 + 7x - 1 = 0}$, we have ${a = -4}$, ${b = 7}$, and ${c = -1}$. Plugging these values into the formula, we get:

${b^2 - 4ac = 7^2 - 4(-4)(-1) = 49 - 16 = 33}$

So, the discriminant is 33.

Number of Real Solutions

The number of real solutions of a quadratic equation depends on the value of the discriminant. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

In our case, the discriminant is 33, which is positive. Therefore, the equation ${-4x^2 + 7x - 1 = 0}$ has two distinct real solutions.

Solving the Equation

Now that we have computed the discriminant and determined the number of real solutions, we can use the quadratic formula to find the solutions. Plugging in the values of ${a}$, ${b}$, and ${c}$, we get:

${x = \frac{-7 \pm \sqrt{33}}{2(-4)} = \frac{-7 \pm \sqrt{33}}{-8}}$

Simplifying, we get two solutions:

${x_1 = \frac{-7 + \sqrt{33}}{-8}}$ ${x_2 = \frac{-7 - \sqrt{33}}{-8}}$

Conclusion

In this article, we have solved the quadratic equation ${-4x^2 + 7x - 1 = 0}$ and computed the value of the discriminant and the number of real solutions. We have used the quadratic formula to find the solutions and have determined that the equation has two distinct real solutions. The discriminant is a crucial concept in solving quadratic equations, and it plays a vital role in determining the number of real solutions.

Applications of Quadratic Equations

Quadratic equations have numerous applications in various fields such as physics, engineering, and economics. Some of the applications include:

• Projectile Motion: Quadratic equations are used to model the trajectory of projectiles under the influence of gravity.
• Optimization: Quadratic equations are used to optimize functions and find the maximum or minimum value of a function.
• Signal Processing: Quadratic equations are used in signal processing to filter signals and remove noise.
• Economics: Quadratic equations are used in economics to model the behavior of economic systems and make predictions about future trends.

Real-World Examples

Quadratic equations have numerous real-world examples, including:

• Designing a Trajectory: A company wants to design a trajectory for a rocket to reach a specific orbit. They use quadratic equations to model the trajectory and ensure that the rocket reaches the desired orbit.
• Optimizing a Function: A company wants to optimize a function to maximize profits. They use quadratic equations to find the maximum value of the function and make decisions about production and pricing.
• Filtering Signals: A company wants to filter signals to remove noise and improve the quality of the signal. They use quadratic equations to design a filter that removes noise and preserves the signal.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. The discriminant is a crucial concept in solving quadratic equations, and it plays a vital role in determining the number of real solutions. We have used the quadratic formula to find the solutions and have determined that the equation has two distinct real solutions. Quadratic equations have numerous real-world examples, and they are used in various fields such as physics, engineering, and economics.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In our previous article, we discussed how to solve quadratic equations and compute the value of the discriminant and the number of real solutions. In this article, we will provide a comprehensive guide to quadratic equations, including a Q&A section to help you understand the concept better.

Quadratic Equation Basics

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.

Quadratic Formula

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

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

This formula is derived from the fact that the quadratic equation can be factored as ${(x - r_1)(x - r_2) = 0}$, where ${r_1}$ and ${r_2}$ are the roots of the equation.

Q&A Section

Q: What is the discriminant, and how is it used in solving quadratic equations?

A: The discriminant is the expression under the square root in the quadratic formula, which is ${b^2 - 4ac}$. It is used to determine the number of real solutions of a quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: How do I determine the number of real solutions of a quadratic equation?

A: To determine the number of real solutions of a quadratic equation, you need to compute the discriminant. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a highest power of two, while a linear equation has a highest power of one.

Q: Can I use the quadratic formula to solve a quadratic equation with complex solutions?

A: Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions. However, you need to be careful when dealing with complex numbers, as they can be represented in different forms.

Q: How do I apply the quadratic formula to solve a quadratic equation?

A: To apply the quadratic formula, you need to plug in the values of ${a}$, ${b}$, and ${c}$ into the formula. Then, you need to simplify the expression and find the solutions.

Real-World Applications of Quadratic Equations

Quadratic equations have numerous real-world applications, including:

• Projectile Motion: Quadratic equations are used to model the trajectory of projectiles under the influence of gravity.
• Optimization: Quadratic equations are used to optimize functions and find the maximum or minimum value of a function.
• Signal Processing: Quadratic equations are used in signal processing to filter signals and remove noise.
• Economics: Quadratic equations are used in economics to model the behavior of economic systems and make predictions about future trends.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. The quadratic formula is a powerful tool for solving quadratic equations, and it is used to determine the number of real solutions of a quadratic equation. We hope that this comprehensive guide to quadratic equations has helped you understand the concept better and has provided you with the tools you need to solve quadratic equations.

Additional Resources

If you want to learn more about quadratic equations, we recommend the following resources:

• Textbooks: "Algebra" by Michael Artin, "Calculus" by Michael Spivak
• Online Resources: Khan Academy, MIT OpenCourseWare, Wolfram Alpha
• Software: Mathematica, Maple, MATLAB

We hope that this article has been helpful in providing you with a comprehensive guide to quadratic equations. If you have any further questions or need additional resources, please don't hesitate to contact us.