Solve For $x$ In The Equation $3x^2 - 18x + 5 = 47$.A. \$x = 3 \pm \sqrt{23}$[/tex\]B. $x = 3 \pm \sqrt{51}$C. $x = 3 \pm \sqrt{41}$D. \$x = 3 \pm \sqrt{5}$[/tex\]

by ADMIN 174 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 focus on solving a specific quadratic equation, $3x^2 - 18x + 5 = 47$, and explore the different methods and techniques used to find the solutions.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In our given equation, $3x^2 - 18x + 5 = 47$, we can rewrite it in the standard form as $3x^2 - 18x - 42 = 0$.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

In our case, $a = 3$, $b = -18$, and $c = -42$. Plugging these values into the quadratic formula, we get:

x=−(−18)±(−18)2−4(3)(−42)2(3)x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4(3)(-42)}}{2(3)}

Simplifying the expression, we get:

x=18±324+5046x = \frac{18 \pm \sqrt{324 + 504}}{6}

x=18±8286x = \frac{18 \pm \sqrt{828}}{6}

x=18±4⋅2076x = \frac{18 \pm \sqrt{4 \cdot 207}}{6}

x=18±22076x = \frac{18 \pm 2\sqrt{207}}{6}

x=9±2073x = \frac{9 \pm \sqrt{207}}{3}

x=3±2079x = 3 \pm \sqrt{\frac{207}{9}}

x=3±23x = 3 \pm \sqrt{23}

Conclusion

In this article, we solved the quadratic equation $3x^2 - 18x + 5 = 47$ using the quadratic formula. We started by rewriting the equation in the standard form, then applied the quadratic formula to find the solutions. The solutions are given by $x = 3 \pm \sqrt{23}$.

Comparison with Answer Choices

Now, let's compare our solution with the answer choices:

A. $x = 3 \pm \sqrt{23}$ B. $x = 3 \pm \sqrt{51}$ C. $x = 3 \pm \sqrt{41}$ D. $x = 3 \pm \sqrt{5}$

Our solution matches with answer choice A.

Tips and Tricks

When solving quadratic equations, it's essential to:

  • Rewrite the equation in the standard form.
  • Apply the quadratic formula correctly.
  • Simplify the expression to find the solutions.

By following these tips and tricks, you can confidently solve quadratic equations and find the solutions.

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.

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 answer some frequently asked questions about quadratic equations, providing a comprehensive guide to help you understand and solve these equations.

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 (in this case, $x$) is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve quadratic equations, including:

  • Factoring: If the quadratic expression can be factored into the product of two binomials, you can solve the equation by setting each factor equal to zero.
  • Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

  • Graphing: You can also solve quadratic equations by graphing the related function and finding the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{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.
  • Plug these values into the quadratic formula.
  • Simplify the expression to find the solutions.

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

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

  1. Identify the values of $a$, $b$, and $c$ in the quadratic equation.
  2. Plug these values into the quadratic formula.
  3. Simplify the expression to find the solutions.
  4. Check the solutions to ensure they are valid.

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

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

  • Not rewriting the equation in the standard form.
  • Not applying the quadratic formula correctly.
  • Not simplifying the expression to find the solutions.
  • Not checking the solutions to ensure they are valid.

Q: How do I check the solutions to ensure they are valid?

A: To check the solutions, you need to:

  • Plug the solutions back into the original equation to ensure they satisfy the equation.
  • Check that the solutions are real and not complex numbers.
  • Check that the solutions are not repeated.

Conclusion

In this article, we answered some frequently asked questions about quadratic equations, providing a comprehensive guide to help you understand and solve these equations. By following the steps outlined in this article, you can confidently solve quadratic equations and find the solutions.

Additional Resources

For more information on quadratic equations, we recommend the following resources:

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

By following these resources and practicing with examples, you can become proficient in solving quadratic equations and apply them to real-world problems.