Solve The Quadratic Equation By Taking Square Roots. Write Your Answer With No Spaces As $x = A, -a$, Where $a$ Is The Value You Found When Solving. Round To The Nearest Tenth.$2x^2 - 9 = 121$

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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 explore how to solve a quadratic equation by taking square roots. We will use the given equation $2x^2 - 9 = 121$ as an example and walk through the step-by-step process of solving it.

Understanding the Quadratic Equation

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, $2x^2 - 9 = 121$, we can rewrite it in the standard form as $2x^2 - 130 = 0$.

Step 1: Isolate the Variable

To solve the quadratic equation, we need to isolate the variable x. We can start by adding 130 to both sides of the equation to get rid of the constant term.

2x2−130=02x^2 - 130 = 0

2x2=1302x^2 = 130

Step 2: Divide by the Coefficient

Next, we need to divide both sides of the equation by the coefficient of the squared term, which is 2.

x2=1302x^2 = \frac{130}{2}

x2=65x^2 = 65

Step 3: Take the Square Root

Now that we have the squared term isolated, we can take the square root of both sides of the equation to solve for x.

x=±65x = \pm \sqrt{65}

Step 4: Simplify the Square Root

The square root of 65 is an irrational number, which means it cannot be expressed as a finite decimal or fraction. However, we can approximate it to the nearest tenth.

x=±8.06x = \pm 8.06

Conclusion

In this article, we solved the quadratic equation $2x^2 - 9 = 121$ by taking square roots. We started by rewriting the equation in the standard form, isolated the variable x, divided by the coefficient, took the square root, and simplified the result. The final solution is $x = \pm 8.06$.

Final Answer

The final answer is $x = 8.06, -8.06$.

Tips and Tricks

  • When solving quadratic equations, make sure to isolate the variable x and simplify the result.
  • Use the square root property to solve for x.
  • Approximate irrational numbers to the nearest tenth.

Common Mistakes

  • Failing to isolate the variable x.
  • Not simplifying the result.
  • Not using the square root property.

Real-World Applications

Quadratic equations have numerous real-world applications, including:

  • Physics: to describe the motion of objects under the influence of gravity or other forces.
  • Engineering: to design and optimize systems, such as bridges or buildings.
  • Economics: to model the behavior of economic systems and make predictions.

Further Reading

For more information on solving quadratic equations, check out the following resources:

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

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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 can use the following steps:

  1. Rewrite the equation in the standard form $ax^2 + bx + c = 0$.
  2. Isolate the variable x by adding or subtracting the same value from both sides of the equation.
  3. Divide both sides of the equation by the coefficient of the squared term.
  4. Take the square root of both sides of the equation to solve for x.
  5. Simplify the result.

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. The general form of a linear equation is $ax + b = 0$, where a and b are constants. Quadratic equations, on the other hand, have a squared term, which makes them more complex and challenging to solve.

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

A: Yes, you can use a calculator to solve a quadratic equation. Many calculators have a built-in quadratic equation solver that can help you find the solutions. However, it's always a good idea to understand the steps involved in solving a quadratic equation, so you can verify the results 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:

  • Failing to isolate the variable x.
  • Not simplifying the result.
  • Not using the square root property.
  • Making errors when dividing or multiplying both sides of the equation.

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

A: To determine if a quadratic equation has real or complex solutions, you can use the discriminant, which is the expression under the square root in the quadratic formula. 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 a quadratic equation?

A: Yes, you can use the quadratic formula to solve a quadratic equation. The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where a, b, and c are the coefficients of the quadratic equation. This formula can be used to find the solutions of a quadratic equation, but it's often more efficient to use the steps outlined above.

Additional Resources

For more information on quadratic equations, check out the following resources:

  • Khan Academy: Quadratic Equations
  • Mathway: Quadratic Equation Solver
  • Wolfram Alpha: Quadratic Equation Solver
  • MIT OpenCourseWare: Quadratic Equations

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding the steps involved in solving a quadratic equation, you can develop a deeper appreciation for the underlying mathematics and improve your problem-solving skills. Whether you're a student or a professional, quadratic equations are an essential part of mathematics, and mastering them can open up new opportunities and challenges.