Solve For $x$. Write Both Solutions, Separated By A Comma.$2x^2 - 7x - 4 = 0$

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Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand the different methods used to solve them. In this article, we will focus on solving the quadratic equation $2x^2 - 7x - 4 = 0$ using the quadratic formula. The quadratic formula is a powerful tool that can be used to solve any quadratic equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

The Quadratic Formula

The quadratic formula is given by:

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

This formula can be used to solve any quadratic equation of the form $ax^2 + bx + c = 0$. To use the quadratic formula, we need to identify the values of $a$, $b$, and $c$ in the given equation.

Identifying the Values of $a$, $b$, and $c$

In the given equation $2x^2 - 7x - 4 = 0$, we can identify the values of $a$, $b$, and $c$ as follows:

$a = 2$ $b = -7$ $c = -4$

Applying the Quadratic Formula

Now that we have identified the values of $a$, $b$, and $c$, we can apply the quadratic formula to solve the equation.

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

$x = \frac{7 \pm \sqrt{49 + 32}}{4}$

$x = \frac{7 \pm \sqrt{81}}{4}$

$x = \frac{7 \pm 9}{4}$

Solving for $x$

Now that we have simplified the expression, we can solve for $x$ by considering both the positive and negative cases.

Case 1: $x = \frac{7 + 9}{4}$

$x = \frac{16}{4}$

$x = 4$

Case 2: $x = \frac{7 - 9}{4}$

$x = \frac{-2}{4}$

$x = -\frac{1}{2}$

Conclusion

In this article, we have solved the quadratic equation $2x^2 - 7x - 4 = 0$ using the quadratic formula. We have identified the values of $a$, $b$, and $c$ in the given equation and applied the quadratic formula to solve for $x$. We have considered both the positive and negative cases and obtained two solutions for $x$. The solutions are $x = 4$ and $x = -\frac{1}{2}$.

Final Answer

The final answer is: $\boxed{4, -\frac{1}{2}}$

Frequently Asked Questions

• What is the quadratic formula?

The quadratic formula is a powerful tool that can be used to solve any quadratic equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

• How do I apply the quadratic formula?

To apply the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the given equation and plug them into the formula.

• What are the solutions to the equation $2x^2 - 7x - 4 = 0$?

The solutions to the equation $2x^2 - 7x - 4 = 0$ are $x = 4$ and $x = -\frac{1}{2}$.

Additional Resources

• Quadratic Formula Calculator

A quadratic formula calculator is a tool that can be used to solve quadratic equations using the quadratic formula.

• Quadratic Equation Solver

A quadratic equation solver is a tool that can be used to solve quadratic equations using various methods, including the quadratic formula.

• Mathematics Tutorials

Mathematics tutorials are online resources that provide step-by-step instructions and examples on how to solve various mathematical problems, including quadratic equations.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they can be used to model a wide range of real-world problems. However, solving quadratic equations can be challenging, especially for those who are new to the subject. In this article, we will provide answers to some of the most frequently asked questions about quadratic equations.

Q&A

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means that 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.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool that can be used to solve any quadratic equation of the form ax^2 + bx + c = 0. It is given by the equation x = (-b ± √(b^2 - 4ac)) / 2a.

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to identify the values of a, b, and c in the given equation and plug them into the formula. You then simplify the expression and solve for x.

Q: What are the solutions to the equation 2x^2 - 7x - 4 = 0?

A: The solutions to the equation 2x^2 - 7x - 4 = 0 are x = 4 and x = -1/2.

Q: Can I use the quadratic formula to solve any quadratic equation?

A: Yes, the quadratic formula can be used to solve any quadratic equation of the form ax^2 + bx + c = 0.

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, the quadratic formula can be used to solve a quadratic equation with complex solutions. However, you will need to use the imaginary unit i to represent the complex solutions.

Q: What is the discriminant in the quadratic formula?

A: The discriminant is the expression b^2 - 4ac that appears in the quadratic formula. It is used to determine the nature of the solutions to the equation.

Q: Can I use the quadratic formula to solve a quadratic equation with a negative discriminant?

A: Yes, the quadratic formula can be used to solve a quadratic equation with a negative discriminant. However, the solutions will be complex numbers.

Q: What is the significance of the quadratic formula in real-world applications?

A: The quadratic formula has many real-world applications, including physics, engineering, and economics. It can be used to model a wide range of problems, including projectile motion, electrical circuits, and population growth.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and they have many real-world applications. In this article, we have provided answers to some of the most frequently asked questions about quadratic equations, including the quadratic formula, the discriminant, and the significance of the quadratic formula in real-world applications.

Final Answer

The final answer is: $\boxed{Yes, the quadratic formula can be used to solve any quadratic equation of the form ax^2 + bx + c = 0.}$

Frequently Asked Questions

• What is the quadratic formula?

The quadratic formula is a powerful tool that can be used to solve any quadratic equation of the form ax^2 + bx + c = 0.

• How do I apply the quadratic formula?

To apply the quadratic formula, you need to identify the values of a, b, and c in the given equation and plug them into the formula.

• What are the solutions to the equation 2x^2 - 7x - 4 = 0?

The solutions to the equation 2x^2 - 7x - 4 = 0 are x = 4 and x = -1/2.

Additional Resources

• Quadratic Formula Calculator

A quadratic formula calculator is a tool that can be used to solve quadratic equations using the quadratic formula.

• Quadratic Equation Solver

A quadratic equation solver is a tool that can be used to solve quadratic equations using various methods, including the quadratic formula.

• Mathematics Tutorials

Mathematics tutorials are online resources that provide step-by-step instructions and examples on how to solve various mathematical problems, including quadratic equations.