Using The Quadratic Formula To Solve $x^2 = 5 - X$, What Are The Values Of $x$?A. $\frac{-1 \pm \sqrt{21}}{2}$ B. $\frac{-1 \pm \sqrt{19}i}{2}$ C. $\frac{5 \pm \sqrt{21}}{2}$ D. $\frac{1 \pm

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Introduction

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Quadratic equations are a fundamental concept in algebra, and solving them is a crucial skill for students to master. In this article, we will explore the quadratic formula and its application to solve quadratic equations. We will use the equation $x^2 = 5 - x$ as a case study to demonstrate the step-by-step process of solving quadratic equations using the quadratic formula.

What is the Quadratic Formula?

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The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$. The quadratic 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 Quadratic Equation $x^2 = 5 - x$

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To solve the quadratic equation $x^2 = 5 - x$, we need to rewrite it in the standard form $ax^2 + bx + c = 0$. We can do this by subtracting $5$ from both sides of the equation:

$$x^2 + x - 5 = 0$$

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

$$a = 1, b = 1, c = -5$$

Applying the Quadratic Formula

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Now that we have identified the coefficients $a$, $b$, and $c$, we can apply the quadratic formula to solve the quadratic equation:

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

Substituting the values of $a$, $b$, and $c$, we get:

$$x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-5)}}{2(1)}$$

Simplifying the expression under the square root, we get:

$$x = \frac{-1 \pm \sqrt{1 + 20}}{2}$$

$$x = \frac{-1 \pm \sqrt{21}}{2}$$

Evaluating the Solutions

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The quadratic formula provides two solutions for the quadratic equation:

$$x = \frac{-1 + \sqrt{21}}{2}$$

$$x = \frac{-1 - \sqrt{21}}{2}$$

Conclusion

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In this article, we have demonstrated the step-by-step process of solving quadratic equations using the quadratic formula. We have used the equation $x^2 = 5 - x$ as a case study to illustrate the application of the quadratic formula. The solutions to the quadratic equation are given by:

$$x = \frac{-1 \pm \sqrt{21}}{2}$$

This is the correct answer to the problem.

Discussion

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The quadratic formula is a powerful tool for solving quadratic equations. However, it is essential to note that the quadratic formula only provides the solutions to the quadratic equation and does not provide any information about the nature of the solutions. In some cases, the solutions may be real and distinct, while in other cases, the solutions may be complex or repeated.

Final Thoughts

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Solving quadratic equations using the quadratic formula requires a clear understanding of the mathematical concepts involved. It is essential to follow the step-by-step process outlined in this article to ensure accurate solutions. With practice and patience, students can master the art of solving quadratic equations using the quadratic formula.

Common Mistakes to Avoid

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When solving quadratic equations using the quadratic formula, students often make mistakes by:

• Not identifying the coefficients $a$, $b$, and $c$ correctly
• Not simplifying the expression under the square root correctly
• Not evaluating the solutions correctly

To avoid these mistakes, students should carefully follow the step-by-step process outlined in this article and double-check their work.

Real-World Applications

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Quadratic equations have numerous real-world applications in fields such as physics, engineering, and economics. For example, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic equation. Similarly, the motion of a pendulum can be described using a quadratic equation.

Conclusion

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In conclusion, solving quadratic equations using the quadratic formula is a crucial skill for students to master. By following the step-by-step process outlined in this article, students can accurately solve quadratic equations and apply the quadratic formula to real-world problems.

Final Thoughts

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The quadratic formula is a powerful tool for solving quadratic equations. With practice and patience, students can master the art of solving quadratic equations using the quadratic formula.

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Introduction

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The quadratic formula is a fundamental concept in algebra, and solving quadratic equations is a crucial skill for students to master. In this article, we will address some of the most frequently asked questions about the quadratic formula and provide clear and concise answers.

Q: What is the quadratic formula?

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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 quadratic formula is given by:

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

Q: How do I apply the quadratic formula?

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A: To apply the quadratic formula, you need to identify the coefficients $a$, $b$, and $c$ in the quadratic equation. Then, you can substitute these values into the quadratic formula and simplify the expression to find the solutions.

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

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

1. Rewrite the quadratic equation in the standard form $ax^2 + bx + c = 0$.
2. Identify the coefficients $a$, $b$, and $c$.
3. Substitute the values of $a$, $b$, and $c$ into the quadratic formula.
4. Simplify the expression under the square root.
5. Evaluate the solutions.

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

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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?

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A: You should use the quadratic formula when:

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

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

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

• Not identifying the coefficients $a$, $b$, and $c$ correctly.
• Not simplifying the expression under the square root correctly.
• Not evaluating the solutions correctly.

Q: How do I check my work when using the quadratic formula?

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A: To check your work when using the quadratic formula, you can:

• Plug the solutions back into the original quadratic equation to verify that they are true.
• Use a calculator to check the solutions.
• Graph the quadratic equation to visualize the solutions.

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

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A: The quadratic formula has numerous real-world applications in fields such as physics, engineering, and economics. For example, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic equation. Similarly, the motion of a pendulum can be described using a quadratic equation.

Conclusion

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In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. By understanding the steps to apply the quadratic formula and avoiding common mistakes, students can accurately solve quadratic equations and apply the quadratic formula to real-world problems.

Final Thoughts

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The quadratic formula is a fundamental concept in algebra, and solving quadratic equations is a crucial skill for students to master. With practice and patience, students can master the art of solving quadratic equations using the quadratic formula.

Additional Resources

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For further learning and practice, students can use online resources such as:

• Khan Academy: Quadratic Formula
• Mathway: Quadratic Formula
• Wolfram Alpha: Quadratic Formula

Conclusion

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In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. By understanding the steps to apply the quadratic formula and avoiding common mistakes, students can accurately solve quadratic equations and apply the quadratic formula to real-world problems.