Use The Quadratic Formula To Solve The Equation X 2 − 2 X = − 9 X^2 - 2x = -9 X 2 − 2 X = − 9 .A. X = − 1 + 2 I 2 X = -1 + 2i\sqrt{2} X = − 1 + 2 I 2 ​ Or X = − 1 − 2 I 2 X = -1 - 2i\sqrt{2} X = − 1 − 2 I 2 ​ B. X = 1 + 10 X = 1 + \sqrt{10} X = 1 + 10 ​ Or X = 1 − 10 X = 1 - \sqrt{10} X = 1 − 10 ​ C. X = 1 + 2 I 2 X = 1 + 2i\sqrt{2} X = 1 + 2 I 2 ​ Or $x = 1 -

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. The quadratic formula is a powerful tool for solving quadratic equations, and in this article, we will explore how to use it to solve the equation $x^2 - 2x = -9$.

What is the Quadratic Formula?

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$. The formula is given by:

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

How to Use the Quadratic Formula

To use the quadratic formula, we need to identify the values of $a$, $b$, and $c$ in the quadratic equation. In this case, we have:

$x^2 - 2x = -9$

We can rewrite this equation as:

$x^2 - 2x + 9 = 0$

Now, we can identify the values of $a$, $b$, and $c$:

$a = 1$ $b = -2$ $c = 9$

Applying the Quadratic Formula

Now that we have identified the values of $a$, $b$, and $c$, we can apply the quadratic formula:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(9)}}{2(1)}$

Simplifying the expression, we get:

$x = \frac{2 \pm \sqrt{4 - 36}}{2}$

$x = \frac{2 \pm \sqrt{-32}}{2}$

$x = \frac{2 \pm 4i\sqrt{2}}{2}$

$x = 1 \pm 2i\sqrt{2}$

Conclusion

In this article, we have used the quadratic formula to solve the equation $x^2 - 2x = -9$. We have identified the values of $a$, $b$, and $c$, and applied the quadratic formula to find the solutions. The solutions are given by:

$x = 1 + 2i\sqrt{2}$ or $x = 1 - 2i\sqrt{2}$

Discussion

The quadratic formula is a powerful tool for solving quadratic equations, and it is widely used in mathematics and science. However, it can be challenging to apply the formula, especially when dealing with complex numbers. In this case, we have used the quadratic formula to solve the equation $x^2 - 2x = -9$, and we have obtained two complex solutions.

Real-World Applications

The quadratic formula has many real-world applications, including:

• Physics: The quadratic formula is used to describe the motion of objects under the influence of gravity.
• Engineering: The quadratic formula is used to design and optimize systems, such as bridges and buildings.
• Computer Science: The quadratic formula is used in algorithms for solving systems of linear equations.

Conclusion

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Frequently Asked Questions

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$. The formula is given by:

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

Q: How do I use the quadratic formula?

A: To use 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 and simplify 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:

1. Identify the values of $a$, $b$, and $c$ in the quadratic equation.
2. Plug these values into the quadratic formula.
3. Simplify the expression to find the solutions.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula and factoring are two different methods for solving quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while the quadratic formula involves using a formula to find the solutions.

Q: When should I use the quadratic formula?

A: You should use the quadratic formula when:

• The quadratic equation cannot be factored easily.
• The quadratic equation has complex solutions.
• You need to find the solutions to a quadratic equation quickly.

Q: What are some common mistakes to avoid when using the quadratic formula?

A: Some common mistakes to avoid when using the quadratic formula include:

• Not identifying the values of $a$, $b$, and $c$ correctly.
• Not simplifying the expression correctly.
• Not checking for complex solutions.

Q: Can I use the quadratic formula to solve quadratic inequalities?

A: No, the quadratic formula is only used to solve quadratic equations, not quadratic inequalities.

Q: Can I use the quadratic formula to solve systems of linear equations?

A: No, the quadratic formula is only used to solve quadratic equations, not systems of linear equations.

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

A: Some real-world applications of the quadratic formula include:

• Physics: The quadratic formula is used to describe the motion of objects under the influence of gravity.
• Engineering: The quadratic formula is used to design and optimize systems, such as bridges and buildings.
• Computer Science: The quadratic formula is used in algorithms for solving systems of linear equations.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations, and it has many real-world applications. We hope that this Q&A article has provided a clear and concise explanation of how to use the quadratic formula, and we encourage readers to try it out for themselves.