Incorrectly simplifying the solution

Oct 13, 2025 by miodeya.com 126 views

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 explore the quadratic formula, a powerful tool for solving quadratic equations. We will use the quadratic formula to solve the equation $x^2 + 9x + 2 = 0$ and provide a step-by-step guide on how to simplify the solution.

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}

where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation.

Solving the Equation $x^2 + 9x + 2 = 0$

To solve the equation $x^2 + 9x + 2 = 0$ , we can use the quadratic formula. We have $a = 1$ , $b = 9$ , and $c = 2$ . Plugging these values into the quadratic formula, we get:

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

Simplifying the expression under the square root, we get:

x = \frac{-9 \pm \sqrt{81 - 8}}{2}

x = \frac{-9 \pm \sqrt{73}}{2}

Simplifying the Solution

The solution to the equation $x^2 + 9x + 2 = 0$ is given by:

x = \frac{-9 \pm \sqrt{73}}{2}

To simplify the solution, we can rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of $\frac{-9 \pm \sqrt{73}}{2}$ is $\frac{-9 \mp \sqrt{73}}{2}$ .

Multiplying the numerator and denominator by the conjugate, we get:

x = \frac{(-9 \pm \sqrt{73})(-9 \mp \sqrt{73})}{2(-9 \pm \sqrt{73})}

Simplifying the expression, we get:

x = \frac{81 \mp 9\sqrt{73} \pm 9\sqrt{73} - 73}{-18 \pm 2\sqrt{73}}

x = \frac{8 \mp 2\sqrt{73}}{-18 \pm 2\sqrt{73}}

Conclusion

In this article, we used the quadratic formula to solve the equation $x^2 + 9x + 2 = 0$ . We simplified the solution by rationalizing the denominator and provided a step-by-step guide on how to simplify the solution. The solution to the equation is given by:

x = \frac{-9 \pm \sqrt{73}}{2}

Real-World Applications

Quadratic equations have numerous real-world applications, including:

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

Common Mistakes

When solving quadratic equations using the quadratic formula, it is common to make mistakes. Some common mistakes include:

Incorrectly simplifying the solution: Make sure to simplify the solution correctly by rationalizing the denominator.
Forgetting to check the solutions: Make sure to check the solutions to ensure that they are valid.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. By using the quadratic formula, we can solve equations of the form $ax^2 + bx + c = 0$ . In this article, we used the quadratic formula to solve the equation $x^2 + 9x + 2 = 0$ and provided a step-by-step guide on how to simplify the solution. The solution to the equation is given by:

x = \frac{-9 \pm \sqrt{73}}{2}

Final Answer

$The final answer is $\boxed{\frac{-9 \pm \sqrt{73}}{2}}$.<br/> **Quadratic Formula Q&A**$

Introduction

The quadratic formula is a powerful tool for solving quadratic equations. In this article, we will answer some of the most frequently asked questions about the quadratic formula.

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, plug these values into the quadratic formula and simplify the expression.

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: Can I use the quadratic formula to solve all quadratic equations?

A: Yes, the quadratic formula can be used to solve all quadratic equations. However, it is not always the most efficient method, especially for simple equations that can be factored easily.

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula, which is given by $b^2 - 4ac$ . The discriminant determines the nature of the solutions to the 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: Can I use the quadratic formula to solve quadratic equations with complex coefficients?

A: Yes, the quadratic formula can be used to solve quadratic equations with complex coefficients. However, the solutions will be complex numbers.

Q: How do I simplify the solutions obtained from the quadratic formula?

A: To simplify the solutions obtained from the quadratic formula, you need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

Q: Can I use the quadratic formula to solve quadratic equations with fractional coefficients?

A: Yes, the quadratic formula can be used to solve quadratic equations with fractional coefficients. However, you need to be careful when simplifying the solutions.

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:

Incorrectly simplifying the solution: Make sure to simplify the solution correctly by rationalizing the denominator.
Forgetting to check the solutions: Make sure to check the solutions to ensure that they are valid.
Using the wrong values for $a$ , $b$ , and $c$ : Make sure to use the correct values for $a$ , $b$ , and $c$ in the quadratic formula.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. By understanding the quadratic formula and its applications, you can solve a wide range of quadratic equations. Remember to be careful when simplifying the solutions and to check the solutions to ensure that they are valid.

Final Answer

The final answer is $\boxed{\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}$ .