Solve For The Roots In Simplest Form Using The Quadratic Formula:$\[ 3x^2 + 42x = -171 \\]Provide Your Answers In The Form:$\[ X = \square \\]$\[ X = \square \\]

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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 focus on solving a quadratic equation using the quadratic formula. We will take a step-by-step approach to simplify the equation and find the roots in their simplest form.

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 formula is given by:

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

Solving the Given Quadratic Equation

Let's apply the quadratic formula to the given equation:

3x2+42x=−1713x^2 + 42x = -171

First, we need to rewrite the equation in the standard form ax2+bx+c=0ax^2 + bx + c = 0. We can do this by subtracting −171-171 from both sides of the equation:

3x2+42x+171=03x^2 + 42x + 171 = 0

Now, we can identify the values of aa, bb, and cc:

  • a=3a = 3
  • b=42b = 42
  • c=171c = 171

Applying the Quadratic Formula

Now that we have the values of aa, bb, and cc, we can plug them into the quadratic formula:

x=−42±422−4(3)(171)2(3)x = \frac{-42 \pm \sqrt{42^2 - 4(3)(171)}}{2(3)}

Simplifying the expression under the square root, we get:

x=−42±1764−20526x = \frac{-42 \pm \sqrt{1764 - 2052}}{6}

x=−42±−2886x = \frac{-42 \pm \sqrt{-288}}{6}

Since the square root of a negative number is not a real number, we need to simplify the expression further. We can do this by factoring out the negative sign:

x=−42±i2886x = \frac{-42 \pm i\sqrt{288}}{6}

Simplifying the expression under the square root, we get:

x=−42±i144⋅26x = \frac{-42 \pm i\sqrt{144 \cdot 2}}{6}

x=−42±i1226x = \frac{-42 \pm i12\sqrt{2}}{6}

Simplifying the expression further, we get:

x=−42±12i26x = \frac{-42 \pm 12i\sqrt{2}}{6}

x=−7±2i21x = \frac{-7 \pm 2i\sqrt{2}}{1}

Simplifying the Roots

Now that we have the roots in the form x=−7±2i21x = \frac{-7 \pm 2i\sqrt{2}}{1}, we can simplify them further. We can do this by factoring out the common factor of 22:

x=−7±2i21x = \frac{-7 \pm 2i\sqrt{2}}{1}

x=−7±2i2x = -7 \pm 2i\sqrt{2}

Conclusion

In this article, we solved a quadratic equation using the quadratic formula. We took a step-by-step approach to simplify the equation and find the roots in their simplest form. We identified the values of aa, bb, and cc, plugged them into the quadratic formula, and simplified the expression to find the roots. The final answer is:

x=−7±2i2x = -7 \pm 2i\sqrt{2}

Final Answer

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In our previous article, we solved a quadratic equation using the quadratic formula. In this article, we will provide a Q&A guide to help you understand quadratic equations and their solutions.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. It is typically written in the form ax^2 + bx + c = 0, where a, b, and c are constants.

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. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to identify the values of a, b, and c in the quadratic equation. Then, plug these values into the formula and simplify the expression to find the roots.

Q: What are the roots of a quadratic equation?

A: The roots of a quadratic equation are the values of x that satisfy the equation. In other words, they are the solutions to the equation.

Q: Can a quadratic equation have more than two roots?

A: No, a quadratic equation can have at most two roots. This is because the quadratic formula provides two solutions for the equation.

Q: What happens if the discriminant (b^2 - 4ac) is negative?

A: If the discriminant is negative, the quadratic equation has no real roots. This means that the solutions to the equation are complex numbers.

Q: How do I simplify complex roots?

A: To simplify complex roots, you need to express them in the form a + bi, where a and b are real numbers and i is the imaginary unit. You can do this by factoring out the common factor of 2 or by using the conjugate of the complex number.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

  • Not identifying the values of a, b, and c correctly
  • Not simplifying the expression under the square root correctly
  • Not expressing complex roots in the correct form
  • Not checking for extraneous solutions

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, you need to plug the solutions back into the original equation and check if they satisfy the equation. If they do not satisfy the equation, they are extraneous solutions.

Conclusion

In this article, we provided a Q&A guide to help you understand quadratic equations and their solutions. We covered topics such as the quadratic formula, roots, and complex numbers. We also discussed common mistakes to avoid when solving quadratic equations and how to check for extraneous solutions. By following these guidelines, you can become more confident in your ability to solve quadratic equations.

Final Tips

  • Always identify the values of a, b, and c correctly
  • Simplify the expression under the square root correctly
  • Express complex roots in the correct form
  • Check for extraneous solutions
  • Practice, practice, practice!

Final Answer

The final answer is 0\boxed{0}.