Solve For X X X .Given Equation: X 2 + 6 X + 9 = 12 X^2 + 6x + 9 = 12 X 2 + 6 X + 9 = 12 Possible Solutions:A. X = − 3 ± 2 3 X = -3 \pm 2 \sqrt{3} X = − 3 ± 2 3 B. X = − 9 ± 4 3 X = -9 \pm 4 \sqrt{3} X = − 9 ± 4 3 C. X = 6 ± 4 3 X = 6 \pm 4 \sqrt{3} X = 6 ± 4 3 D. X = 3 ± 2 3 X = 3 \pm 2 \sqrt{3} X = 3 ± 2 3

Mar 12, 2025 by ADMIN 318 views

**Solving Quadratic Equations: A Step-by-Step Guide**

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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, $x^2 + 6x + 9 = 12$ , and explore the possible solutions. We will break down the solution process into manageable steps, making it easier to understand and apply.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$ ) is two. The general form of a quadratic equation is:

ax^2 + bx + c = 0

where $a$ , $b$ , and $c$ are constants. In our given equation, $x^2 + 6x + 9 = 12$ , we can rewrite it in the standard form by subtracting 12 from both sides:

x^2 + 6x - 3 = 0

The Quadratic Formula

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

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

In our equation, $a = 1$ , $b = 6$ , and $c = -3$ . Plugging these values into the quadratic formula, we get:

x = \frac{-6 \pm \sqrt{6^2 - 4(1)(-3)}}{2(1)}

Simplifying the Quadratic Formula

Now, let's simplify the expression under the square root:

6^2 - 4(1)(-3) = 36 + 12 = 48

So, the quadratic formula becomes:

x = \frac{-6 \pm \sqrt{48}}{2}

Simplifying the Square Root

The square root of 48 can be simplified by factoring out a perfect square:

\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}

Now, we can rewrite the quadratic formula as:

x = \frac{-6 \pm 4\sqrt{3}}{2}

Simplifying the Expression

To simplify the expression further, we can divide both the numerator and the denominator by 2:

x = -3 \pm 2\sqrt{3}

Conclusion

In this article, we solved the quadratic equation $x^2 + 6x + 9 = 12$ using the quadratic formula. We broke down the solution process into manageable steps, making it easier to understand and apply. The possible solutions are:

$x = -3 \pm 2\sqrt{3}$

This solution matches option A in the given problem.

Final Thoughts

Solving quadratic equations is a crucial skill for students and professionals alike. By understanding the quadratic formula and applying it to specific equations, we can find the possible solutions. In this article, we focused on solving a specific quadratic equation and explored the possible solutions. We hope this article has provided valuable insights and helped readers understand the solution process.

Common Mistakes to Avoid

When solving quadratic equations, it's essential to avoid common mistakes. Here are a few:

Incorrectly applying the quadratic formula: Make sure to plug in the correct values for $a$ , $b$ , and $c$ .
Not simplifying the expression: Take the time to simplify the expression under the square root and the final solution.
Not checking the solutions: Verify that the solutions satisfy the original equation.

By avoiding these common mistakes, you can ensure that your solutions are accurate and reliable.

Real-World Applications

Quadratic equations have numerous real-world applications. Here are a few examples:

Physics: Quadratic equations are used to model the motion of objects under the influence of gravity or other forces.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges or buildings.
Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

By understanding quadratic equations and their applications, you can develop a deeper appreciation for the mathematical concepts and their real-world implications.

Conclusion

In conclusion, solving quadratic equations is a crucial skill for students and professionals alike. By understanding the quadratic formula and applying it to specific equations, we can find the possible solutions. In this article, we focused on solving a specific quadratic equation and explored the possible solutions. We hope this article has provided valuable insights and helped readers understand the solution process.

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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 our previous article, we explored the solution process for a specific quadratic equation, $x^2 + 6x + 9 = 12$ . In this article, we will answer some frequently asked questions about quadratic equations and provide additional insights and tips.

Q&A

Q: What is a quadratic equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$ ) is two. The general form of a quadratic equation is:

ax^2 + bx + c = 0

Q: How do I solve a quadratic equation?

To solve a quadratic equation, you can use the quadratic formula:

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

Alternatively, you can factor the equation or complete the square.

Q: What is the quadratic formula?

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

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

Q: How do I simplify the quadratic formula?

To simplify the quadratic formula, you can start by simplifying the expression under the square root. This may involve factoring or using the difference of squares formula.

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

Some common mistakes to avoid when solving quadratic equations include:

Incorrectly applying the quadratic formula: Make sure to plug in the correct values for $a$ , $b$ , and $c$ .
Not simplifying the expression: Take the time to simplify the expression under the square root and the final solution.
Not checking the solutions: Verify that the solutions satisfy the original equation.

Q: How do I check my solutions?

To check your solutions, you can plug them back into the original equation and verify that they satisfy the equation. You can also use the quadratic formula to check your solutions.

Q: What are some real-world applications of quadratic equations?

Quadratic equations have numerous real-world applications, including:

Physics: Quadratic equations are used to model the motion of objects under the influence of gravity or other forces.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges or buildings.
Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Q: How do I choose between factoring and the quadratic formula?

When deciding between factoring and the quadratic formula, consider the following:

Factoring: If the equation can be easily factored, it may be faster and more efficient to use factoring.
Quadratic formula: If the equation cannot be easily factored, or if you are unsure about how to factor it, the quadratic formula may be a better choice.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding the quadratic formula and applying it to specific equations, we can find the possible solutions. In this article, we answered some frequently asked questions about quadratic equations and provided additional insights and tips. We hope this article has provided valuable insights and helped readers understand the solution process.

Final Thoughts

Solving quadratic equations is a crucial skill for students and professionals alike. By understanding the quadratic formula and applying it to specific equations, we can find the possible solutions. In this article, we focused on answering some frequently asked questions about quadratic equations and provided additional insights and tips. We hope this article has provided valuable insights and helped readers understand the solution process.

Additional Resources

For additional resources on quadratic equations, including practice problems and video tutorials, check out the following:

Khan Academy: Khan Academy offers a comprehensive course on quadratic equations, including video tutorials and practice problems.
Mathway: Mathway is an online math problem solver that can help you solve quadratic equations and other math problems.
Wolfram Alpha: Wolfram Alpha is a powerful online calculator that can help you solve quadratic equations and other math problems.

By using these resources and practicing regularly, you can develop a deeper understanding of quadratic equations and improve your problem-solving skills.