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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. One method of solving quadratic equations is by completing the square. This method involves manipulating the equation to create a perfect square trinomial, which can then be factored to find the solutions. In this article, we will explore how to solve quadratic equations by completing the square, using the equation -2g^2 - 12g + 14 = 0 as an example.

What is Completing the Square?

Completing the square is a technique used to solve quadratic equations by manipulating the equation to create a perfect square trinomial. This is done by adding and subtracting a constant term to the equation, which allows us to factor the equation into a product of two binomials. The constant term added is the square of half the coefficient of the linear term.

Step 1: Write the Equation in Standard Form

The first step in solving a quadratic equation by completing the square is to write the equation in standard form. The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants. In our example equation, -2g^2 - 12g + 14 = 0, we can rewrite it as -2g^2 - 12g = -14.

Step 2: Move the Constant Term to the Right Side

Next, we need to move the constant term to the right side of the equation. This is done by adding 14 to both sides of the equation, resulting in -2g^2 - 12g + 14 = 0 becoming -2g^2 - 12g = -14 + 14.

Step 3: Factor Out the Coefficient of the Quadratic Term

Now, we need to factor out the coefficient of the quadratic term, which is -2. This is done by dividing both sides of the equation by -2, resulting in g^2 + 6g = 7.

Step 4: Add and Subtract the Constant Term

The next step is to add and subtract the constant term, which is the square of half the coefficient of the linear term. In this case, the coefficient of the linear term is 6, so half of it is 3. The square of 3 is 9, so we add and subtract 9 to the equation, resulting in g^2 + 6g + 9 = 7 + 9.

Step 5: Factor the Equation

Now that we have added and subtracted the constant term, we can factor the equation. The equation g^2 + 6g + 9 = 16 can be factored as (g + 3)^2 = 16.

Step 6: Solve for g

Finally, we can solve for g by taking the square root of both sides of the equation. This results in g + 3 = ±√16, which simplifies to g + 3 = ±4.

Solving for g

To solve for g, we need to isolate g on one side of the equation. This is done by subtracting 3 from both sides of the equation, resulting in g = -3 ± 4.

Simplifying the Solutions

The solutions to the equation g = -3 ± 4 can be simplified by combining the constants. This results in two possible solutions: g = -3 + 4 = 1 and g = -3 - 4 = -7.

Conclusion

In this article, we have explored how to solve quadratic equations by completing the square. We used the equation -2g^2 - 12g + 14 = 0 as an example and walked through the steps of completing the square, including writing the equation in standard form, moving the constant term to the right side, factoring out the coefficient of the quadratic term, adding and subtracting the constant term, factoring the equation, and solving for g. The solutions to the equation were g = 1 and g = -7.

Example Problems

Here are a few example problems to help you practice solving quadratic equations by completing the square:

  • Solve the equation x^2 + 5x + 6 = 0 by completing the square.
  • Solve the equation y^2 - 3y - 4 = 0 by completing the square.
  • Solve the equation z^2 + 2z + 1 = 0 by completing the square.

Tips and Tricks

Here are a few tips and tricks to help you solve quadratic equations by completing the square:

  • Make sure to write the equation in standard form before starting the process.
  • Be careful when adding and subtracting the constant term, as this can affect the solutions.
  • Use the correct formula for factoring the equation, which is (g + a)^2 = b^2.
  • Check your solutions by plugging them back into the original equation.

Common Mistakes

Here are a few common mistakes to avoid when solving quadratic equations by completing the square:

  • Failing to write the equation in standard form.
  • Adding and subtracting the constant term incorrectly.
  • Factoring the equation incorrectly.
  • Not checking the solutions.

Real-World Applications

Quadratic equations have many 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 the behavior of economic systems, such as supply and demand.
  • Computer Science: Quadratic equations are used in algorithms and data structures, such as sorting and searching.

Conclusion

Frequently Asked Questions

Q: What is completing the square?

A: Completing the square is a technique used to solve quadratic equations by manipulating the equation to create a perfect square trinomial. This is done by adding and subtracting a constant term to the equation, which allows us to factor the equation into a product of two binomials.

Q: How do I know when to use completing the square?

A: You should use completing the square when the quadratic equation is in the form ax^2 + bx + c = 0, where a, b, and c are constants. This method is particularly useful when the equation is not easily factorable.

Q: What are the steps to complete the square?

A: The steps to complete the square are:

  1. Write the equation in standard form.
  2. Move the constant term to the right side of the equation.
  3. Factor out the coefficient of the quadratic term.
  4. Add and subtract the constant term.
  5. Factor the equation.
  6. Solve for the variable.

Q: How do I add and subtract the constant term?

A: To add and subtract the constant term, you need to find the square of half the coefficient of the linear term. This is done by taking half of the coefficient of the linear term and squaring it. You then add and subtract this value to the equation.

Q: What if I get a negative value when adding and subtracting the constant term?

A: If you get a negative value when adding and subtracting the constant term, you need to add the absolute value of the negative value to both sides of the equation. This will ensure that the equation remains balanced.

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

A: Yes, you can use completing the square to solve quadratic equations with complex coefficients. However, you need to be careful when working with complex numbers and ensure that you follow the correct procedures.

Q: How do I check my solutions?

A: To check your solutions, you need to plug them back into the original equation and ensure that they satisfy the equation. This will help you verify that your solutions are correct.

Q: What are some common mistakes to avoid when completing the square?

A: Some common mistakes to avoid when completing the square include:

  • Failing to write the equation in standard form.
  • Adding and subtracting the constant term incorrectly.
  • Factoring the equation incorrectly.
  • Not checking the solutions.

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

A: Yes, you can use completing the square to solve quadratic equations with rational coefficients. However, you need to be careful when working with rational numbers and ensure that you follow the correct procedures.

Q: How do I apply completing the square to real-world problems?

A: Completing the square can be applied to a wide range of real-world problems, including physics, engineering, economics, and computer science. You can use this technique to model and solve problems involving quadratic equations.

Q: What are some tips and tricks for completing the square?

A: Some tips and tricks for completing the square include:

  • Make sure to write the equation in standard form before starting the process.
  • Be careful when adding and subtracting the constant term, as this can affect the solutions.
  • Use the correct formula for factoring the equation, which is (g + a)^2 = b^2.
  • Check your solutions by plugging them back into the original equation.

Conclusion

In conclusion, completing the square is a powerful technique for solving quadratic equations. By following the steps outlined in this article and avoiding common mistakes, you can master this technique and apply it to a wide range of problems. Remember to practice regularly and to check your solutions to ensure accuracy.