Using The Quadratic Formula To Solve $x^2 + 20 = 2x$, What Are The Values Of $x$?A. $1 \pm \sqrt{21} I$ B. $-1 \pm \sqrt{19} I$ C. $1 + 2\sqrt{19} J$ D. $1 \pm \sqrt{19}$

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Introduction

Quadratic equations are a fundamental concept in mathematics, and they can be solved using various methods, including factoring, completing the square, and the quadratic formula. In this article, we will focus on using the quadratic formula to solve a quadratic equation, and we will explore the different steps involved in this process.

What is the Quadratic Formula?

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation of the form ax2+bx+c=0ax^2 + bx + c = 0. The quadratic formula is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula is used to find the values of xx that satisfy the quadratic equation.

Step 1: Identify the Coefficients

To use the quadratic formula, we need to identify the coefficients aa, bb, and cc in the quadratic equation. In the equation x2+20=2xx^2 + 20 = 2x, we can rewrite it as x2−2x+20=0x^2 - 2x + 20 = 0. Comparing this equation with the general form ax2+bx+c=0ax^2 + bx + c = 0, we can see that a=1a = 1, b=−2b = -2, and c=20c = 20.

Step 2: Plug in the Coefficients

Now that we have identified the coefficients, we can plug them into the quadratic formula:

x=−(−2)±(−2)2−4(1)(20)2(1)x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(20)}}{2(1)}

Step 3: Simplify the Expression

Simplifying the expression inside the square root, we get:

x=2±4−802x = \frac{2 \pm \sqrt{4 - 80}}{2}

x=2±−762x = \frac{2 \pm \sqrt{-76}}{2}

Step 4: Simplify the Square Root

To simplify the square root, we can rewrite it as:

x=2±−4⋅192x = \frac{2 \pm \sqrt{-4 \cdot 19}}{2}

x=2±2i192x = \frac{2 \pm 2i\sqrt{19}}{2}

Step 5: Simplify the Expression

Simplifying the expression, we get:

x=1±i19x = 1 \pm i\sqrt{19}

Conclusion

Using the quadratic formula, we have solved the quadratic equation x2+20=2xx^2 + 20 = 2x and found the values of xx to be 1±i191 \pm i\sqrt{19}. This solution is in the form of complex numbers, which is a common outcome when solving quadratic equations using the quadratic formula.

Comparison with Answer Choices

Comparing our solution with the answer choices, we can see that our solution matches with answer choice A: 1±21i1 \pm \sqrt{21} i. However, we can see that our solution is actually 1±i191 \pm i\sqrt{19}, which is different from the answer choice.

Discussion

The quadratic formula is a powerful tool for solving quadratic equations, but it can be challenging to use, especially when dealing with complex numbers. In this article, we have walked through the steps involved in using the quadratic formula to solve a quadratic equation, and we have seen how to simplify the expression inside the square root.

Tips and Tricks

When using the quadratic formula, it's essential to simplify the expression inside the square root as much as possible. This can involve factoring the expression or rewriting it in a different form. Additionally, when dealing with complex numbers, it's crucial to remember that i2=−1i^2 = -1.

Real-World Applications

The quadratic formula has numerous real-world applications, including physics, engineering, and economics. For example, in physics, the quadratic formula is used to describe the motion of objects under the influence of gravity or other forces. In engineering, the quadratic formula is used to design and optimize systems, such as bridges or buildings.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations, and it has numerous real-world applications. By following the steps outlined in this article, we can use the quadratic formula to solve quadratic equations and find the values of xx that satisfy the equation.

Final Answer

The final answer is: 1±i19\boxed{1 \pm i\sqrt{19}}

Introduction

The quadratic formula is a powerful tool for solving quadratic equations, but it can be challenging to use, especially for those who are new to it. In this article, we will answer some of the most frequently asked questions about the quadratic formula, including how to use it, common mistakes to avoid, and real-world applications.

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 ax2+bx+c=0ax^2 + bx + c = 0. The quadratic formula is given by:

x=−b±b2−4ac2ax = \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 coefficients aa, bb, and cc in the quadratic equation. Then, you can plug these values into the quadratic formula and simplify the expression inside the square root.

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 finding two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term. The quadratic formula, on the other hand, involves using a formula to find the solutions to the quadratic equation.

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 coefficients aa, bb, and cc correctly
  • Not simplifying the expression inside the square root correctly
  • Not using the correct formula for complex numbers

Q: Can the quadratic formula be used to solve all types of quadratic equations?

A: No, the quadratic formula can only be used to solve quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0. It cannot be used to solve quadratic equations that are not in this form.

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

A: The quadratic formula has numerous real-world applications, including physics, engineering, and economics. For example, in physics, the quadratic formula is used to describe the motion of objects under the influence of gravity or other forces. In engineering, the quadratic formula is used to design and optimize systems, such as bridges or buildings.

Q: Can the quadratic formula be used to solve quadratic equations with complex numbers?

A: Yes, the quadratic formula can be used to solve quadratic equations with complex numbers. When dealing with complex numbers, it's essential to remember that i2=−1i^2 = -1.

Q: How do I know if the quadratic formula will give me real or complex solutions?

A: To determine if the quadratic formula will give you real or complex solutions, you need to look at the expression inside the square root. If the expression is positive, then the quadratic formula will give you real solutions. If the expression is negative, then the quadratic formula will give you complex solutions.

Q: Can the quadratic formula be used to solve quadratic equations with variables in the coefficients?

A: No, the quadratic formula cannot be used to solve quadratic equations with variables in the coefficients. In this case, you need to use a different method, such as substitution or elimination.

Conclusion

In conclusion, the quadratic formula is a powerful tool for solving quadratic equations, and it has numerous real-world applications. By following the steps outlined in this article, you can use the quadratic formula to solve quadratic equations and find the values of xx that satisfy the equation.

Final Answer

The final answer is: 1±i19\boxed{1 \pm i\sqrt{19}}