Solve By The Quadratic Formula. List The Solutions, Separated By Commas.${10x^2 - 2x - 8 = 0}$ {x = \}

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Introduction

The quadratic formula is a powerful tool used to solve quadratic equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. In this article, we will explore how to use the quadratic formula to solve quadratic equations and provide a step-by-step guide on how to list the solutions, separated by commas.

What is the Quadratic Formula?

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

x=−b±b2−4ac2a{ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} }

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

How to Use the Quadratic Formula

To use the quadratic formula, you need to follow these steps:

  1. Identify the coefficients: Identify the coefficients a, b, and c in the quadratic equation.
  2. Plug in the values: Plug in the values of a, b, and c into the quadratic formula.
  3. Simplify the expression: Simplify the expression under the square root.
  4. Solve for x: Solve for x by simplifying the expression.

Step-by-Step Solution

Let's use the quadratic equation 10x^2 - 2x - 8 = 0 as an example.

Step 1: Identify the Coefficients

The coefficients of the quadratic equation are a = 10, b = -2, and c = -8.

Step 2: Plug in the Values

Plug in the values of a, b, and c into the quadratic formula:

x=−(−2)±(−2)2−4(10)(−8)2(10){ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(10)(-8)}}{2(10)} }

Step 3: Simplify the Expression

Simplify the expression under the square root:

x=2±4+32020{ x = \frac{2 \pm \sqrt{4 + 320}}{20} }

x=2±32420{ x = \frac{2 \pm \sqrt{324}}{20} }

x=2±1820{ x = \frac{2 \pm 18}{20} }

Step 4: Solve for x

Solve for x by simplifying the expression:

x=2+1820orx=2−1820{ x = \frac{2 + 18}{20} \quad \text{or} \quad x = \frac{2 - 18}{20} }

x=2020orx=−1620{ x = \frac{20}{20} \quad \text{or} \quad x = \frac{-16}{20} }

x=1orx=−45{ x = 1 \quad \text{or} \quad x = -\frac{4}{5} }

Listing the Solutions

The solutions to the quadratic equation 10x^2 - 2x - 8 = 0 are x = 1 and x = -\frac{4}{5}.

Conclusion

In this article, we have explored how to use the quadratic formula to solve quadratic equations and provided a step-by-step guide on how to list the solutions, separated by commas. We have used the quadratic equation 10x^2 - 2x - 8 = 0 as an example and solved for x using the quadratic formula. The solutions to the quadratic equation are x = 1 and x = -\frac{4}{5}.

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.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the coefficients a, b, and c in the quadratic equation, plug in the values into the formula, simplify the expression under the square root, and solve for x.

Q: What are the solutions to the quadratic equation 10x^2 - 2x - 8 = 0?

A: The solutions to the quadratic equation 10x^2 - 2x - 8 = 0 are x = 1 and x = -\frac{4}{5}.

References

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.

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Introduction

The quadratic formula is a powerful tool used to solve quadratic equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. In this article, we will provide a comprehensive Q&A section on the quadratic formula, covering frequently asked questions and providing detailed answers to help you better understand the concept.

Q&A Section

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

x=−b±b2−4ac2a{ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} }

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

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to follow these steps:

  1. Identify the coefficients: Identify the coefficients a, b, and c in the quadratic equation.
  2. Plug in the values: Plug in the values of a, b, and c into the quadratic formula.
  3. Simplify the expression: Simplify the expression under the square root.
  4. Solve for x: Solve for x by simplifying the expression.

Q: What are the solutions to the quadratic equation 10x^2 - 2x - 8 = 0?

A: The solutions to the quadratic equation 10x^2 - 2x - 8 = 0 are x = 1 and x = -\frac{4}{5}.

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

A: Yes, the quadratic formula can be used to solve all types of quadratic equations, including those with complex solutions.

Q: How do I determine if a quadratic equation has real or complex solutions?

A: To determine if a quadratic equation has real or complex solutions, you need to check the discriminant (b^2 - 4ac). If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

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

A: No, the quadratic formula is only applicable to quadratic equations with real coefficients.

Q: How do I simplify the expression under the square root in the quadratic formula?

A: To simplify the expression under the square root, you need to follow these steps:

  1. Simplify the expression: Simplify the expression under the square root.
  2. Check for perfect squares: Check if the expression under the square root is a perfect square.
  3. Simplify the square root: Simplify the square root of the expression.

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

A: Yes, the quadratic formula can be used to solve quadratic equations with rational coefficients.

Q: How do I determine if a quadratic equation has rational or irrational solutions?

A: To determine if a quadratic equation has rational or irrational solutions, you need to check the solutions obtained using the quadratic formula. If the solutions are rational numbers, the equation has rational solutions. If the solutions are irrational numbers, the equation has irrational solutions.

Conclusion

In this article, we have provided a comprehensive Q&A section on the quadratic formula, covering frequently asked questions and providing detailed answers to help you better understand the concept. We hope that this article has been helpful in clarifying any doubts you may have had about the quadratic formula.

Frequently Asked Questions (FAQs)

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to follow these steps:

  1. Identify the coefficients: Identify the coefficients a, b, and c in the quadratic equation.
  2. Plug in the values: Plug in the values of a, b, and c into the quadratic formula.
  3. Simplify the expression: Simplify the expression under the square root.
  4. Solve for x: Solve for x by simplifying the expression.

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

A: Yes, the quadratic formula can be used to solve all types of quadratic equations, including those with complex solutions.

References

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.