Solve $x^2 - 4 = 77$.A. $x = \sqrt{73}$ Or $ X = − 73 X = -\sqrt{73} X = − 73 ​ [/tex] B. $x = 9$ Or $x = -9$ C. $ X = 11 X = 11 X = 11 [/tex] Or $x = 7$ D. $x = 9i$ Or $x =

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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 the quadratic equation $x^2 - 4 = 77$. We will break down the solution step by step, using algebraic manipulations and mathematical concepts to arrive at the final answer.

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, $x^2 - 4 = 77$, we can rewrite it in the standard form as $x^2 - 81 = 0$ by adding 81 to both sides.

Step 1: Rearrange the Equation

To solve the equation, we need to isolate the variable x. We can start by rearranging the equation to get all the terms on one side. In this case, we have:

x2−81=0x^2 - 81 = 0

Step 2: Factor the Equation

The next step is to factor the equation, if possible. In this case, we can factor the left-hand side as:

(x−9)(x+9)=0(x - 9)(x + 9) = 0

Step 3: Solve for x

Now that we have factored the equation, we can solve for x by setting each factor equal to zero. This gives us two possible solutions:

x−9=0orx+9=0x - 9 = 0 \quad \text{or} \quad x + 9 = 0

Solving for x in each case, we get:

x=9orx=−9x = 9 \quad \text{or} \quad x = -9

Conclusion

In conclusion, we have solved the quadratic equation $x^2 - 4 = 77$ by rearranging it, factoring it, and solving for x. The final answer is:

x=9orx=−9x = 9 \quad \text{or} \quad x = -9

Comparison with Other Options

Let's compare our solution with the options provided:

A. $x = \sqrt{73}$ or $x = -\sqrt{73}$

B. $x = 9$ or $x = -9$

C. $x = 11$ or $x = 7$

D. $x = 9i$ or $x = -9i$

Our solution matches option B, which is $x = 9$ or $x = -9$.

Why is Our Solution Correct?

Our solution is correct because we followed the correct steps to solve the quadratic equation. We rearranged the equation, factored it, and solved for x. We also compared our solution with the options provided and found that it matches option B.

Common Mistakes to Avoid

When solving quadratic equations, there are several common mistakes to avoid:

  • Not rearranging the equation to get all the terms on one side
  • Not factoring the equation, if possible
  • Not solving for x by setting each factor equal to zero
  • Not comparing the solution with the options provided

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.

Conclusion

In conclusion, solving quadratic equations is a crucial skill for students and professionals alike. By following the correct steps and avoiding common mistakes, we can arrive at the correct solution. In this article, we solved the quadratic equation $x^2 - 4 = 77$ and found that the final answer is:

x=9orx=−9x = 9 \quad \text{or} \quad x = -9

Frequently Asked Questions

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: To solve a quadratic equation, you need to follow these steps:

  1. Rearrange the equation to get all the terms on one side.
  2. Factor the equation, if possible.
  3. Solve for x by setting each factor equal to zero.

Q: What is the difference between a quadratic equation and a linear equation?

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (in this case, x) is one. A quadratic equation, on the other hand, is a polynomial equation of degree two.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. However, it's always a good idea to understand the steps involved in solving a quadratic equation, so you can verify the answer and learn from the process.

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

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

  • Not rearranging the equation to get all the terms on one side
  • Not factoring the equation, if possible
  • Not solving for x by setting each factor equal to zero
  • Not comparing the solution with the options provided

Q: How do I know if a quadratic equation can be factored?

A: A quadratic equation can be factored if it can be written in the form $(x - r)(x - s) = 0$, where r and s are constants. If the equation cannot be factored, you may need to use other methods, such as the quadratic formula.

Q: What is the quadratic formula?

A: The quadratic formula is a method for solving quadratic equations that cannot be factored. It is given by:

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

Q: When should I use the quadratic formula?

A: You should use the quadratic formula when the quadratic equation cannot be factored. This is often the case when the equation has complex roots.

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

A: Yes, you can use the quadratic formula to solve quadratic equations with complex roots. The quadratic formula will give you the complex roots of the equation.

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

A: 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.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By following the correct steps and avoiding common mistakes, we can arrive at the correct solution. We hope this Q&A article has been helpful in answering your questions about quadratic equations.