Solve The Quadratic Equation. Write Your Solutions In The Blanks, Starting With The Smaller Value. 4 X 2 − 7 = 9 4x^2 - 7 = 9 4 X 2 − 7 = 9 Solution: { □ \square □ , □ \square □ }

Mar 13, 2025 by ADMIN 180 views

Solve the Quadratic Equation

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, $4x^2 - 7 = 9$ , and provide step-by-step solutions to find the values of $x$ . We will also discuss the importance of quadratic equations and their applications in various fields.

What are Quadratic Equations?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

This formula can be used to find the solutions to any quadratic equation in the form $ax^2 + bx + c = 0$ .

Solving the Quadratic Equation

Now, let's apply the quadratic formula to solve the equation $4x^2 - 7 = 9$ . First, we need to rewrite the equation in the standard form $ax^2 + bx + c = 0$ . We can do this by subtracting 9 from both sides of the equation:

$4x^2 - 7 - 9 = 0$

Simplifying the equation, we get:

$4x^2 - 16 = 0$

Now, we can use the quadratic formula to find the solutions:

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

In this case, $a = 4$ , $b = 0$ , and $c = -16$ . Plugging these values into the formula, we get:

$x = \frac{-0 \pm \sqrt{0^2 - 4(4)(-16)}}{2(4)}$

Simplifying the expression under the square root, we get:

$x = \frac{\pm \sqrt{256}}{8}$

$x = \frac{\pm 16}{8}$

Simplifying further, we get:

$x = \pm 2$

Therefore, the solutions to the equation $4x^2 - 7 = 9$ are $x = 2$ and $x = -2$ .

Discussion

Quadratic equations are an essential part of mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we have focused on solving a specific quadratic equation, $4x^2 - 7 = 9$ , and provided step-by-step solutions to find the values of $x$ . We have also discussed the importance of quadratic equations and their applications in various fields.

Quadratic equations have numerous applications 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 financial markets.

In conclusion, solving quadratic equations is a fundamental skill that is essential for students and professionals alike. By understanding the concepts and methods of solving quadratic equations, we can apply them to real-world problems and make informed decisions.

Conclusion

In this article, we have solved the quadratic equation $4x^2 - 7 = 9$ using the quadratic formula. We have provided step-by-step solutions to find the values of $x$ and discussed the importance of quadratic equations and their applications in various fields. We hope that this article has provided valuable insights and knowledge on solving quadratic equations.

Final Answer

The final answer to the quadratic equation $4x^2 - 7 = 9$ is:

$x = \boxed{2}$ and $x = \boxed{-2}$

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Introduction

In our previous article, we solved the quadratic equation $4x^2 - 7 = 9$ using the quadratic formula. In this article, we will provide a Q&A section to address common questions and concerns 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. It is typically written in the form $ax^2 + bx + c = 0$ , 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, completing the square, and the quadratic formula. The quadratic formula is a powerful tool that can be used to find the solutions to any quadratic equation in the form $ax^2 + bx + c = 0$ .

Q: What is the quadratic formula?

A: The quadratic formula is given by:

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

This formula can be used to find the solutions to any quadratic equation in the form $ax^2 + bx + c = 0$ .

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 quadratic equation. Then, you can plug these values into the formula to find the solutions.

Q: What are the steps to solve a quadratic equation using the quadratic formula?

A: The steps to solve a quadratic equation using the quadratic formula are:

Identify the values of $a$ , $b$ , and $c$ in the quadratic equation.
Plug these values into the quadratic formula.
Simplify the expression under the square root.
Simplify the expression to find the solutions.

Q: What are the common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

Not identifying the values of $a$ , $b$ , and $c$ correctly.
Not simplifying the expression under the square root correctly.
Not simplifying the expression to find the solutions correctly.

Q: How do I check my solutions?

A: To check your solutions, you can plug the values of $x$ back into the original quadratic equation to see if they satisfy the equation.

Q: What are the applications of quadratic equations?

A: Quadratic equations have numerous applications 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 financial markets.

Q: Can I use quadratic equations to solve real-world problems?

A: Yes, quadratic equations can be used to solve real-world problems. By understanding the concepts and methods of solving quadratic equations, you can apply them to real-world problems and make informed decisions.

Q: Where can I find more information about quadratic equations?

A: You can find more information about quadratic equations in various resources, including textbooks, online tutorials, and educational websites.

Conclusion

In this article, we have provided a Q&A section to address common questions and concerns about quadratic equations. We hope that this article has provided valuable insights and knowledge on quadratic equations.

Final Answer

The final answer to the quadratic equation $4x^2 - 7 = 9$ is:

$x = \boxed{2}$ and $x = \boxed{-2}$