Solve The Equation By Completing The Square. Round To The Nearest Hundredth If Necessary. $\[ X^2 + 3x = 24 \\]A. 3.55, -6.55 B. 3.62, -6.62 C. 4.66, 5.12 D. 24.75, -27.75

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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 of the methods used to solve quadratic equations is by completing the square. This method involves manipulating the equation to express it in the form of a perfect square trinomial, which can then be easily solved. In this article, we will explore how to solve quadratic equations by completing the square and provide examples to illustrate the process.

What is Completing the Square?

Completing the square is a technique used to solve quadratic equations of the form ax^2 + bx + c = 0. The method involves manipulating the equation to express it in the form of a perfect square trinomial, which can then be easily solved. The perfect square trinomial has the form (x + d)^2, where d is a constant.

Step-by-Step Guide to Completing the Square

To complete the square, follow these steps:

  1. Write the equation in the form ax^2 + bx + c = 0: Start by writing the quadratic equation in the standard form.
  2. Move the constant term to the right-hand side: Move the constant term to the right-hand side of the equation by subtracting it from both sides.
  3. Divide the coefficient of the x-term by 2: Divide the coefficient of the x-term by 2 and square the result.
  4. Add the squared result to both sides: Add the squared result to both sides of the equation.
  5. Factor the left-hand side: Factor the left-hand side of the equation as a perfect square trinomial.
  6. Solve for x: Solve for x by taking the square root of both sides of the equation.

Example 1: Solving x^2 + 3x = 24

Let's use the equation x^2 + 3x = 24 as an example. To solve this equation by completing the square, follow the steps outlined above.

Step 1: Write the equation in the form ax^2 + bx + c = 0

x^2 + 3x = 24

Step 2: Move the constant term to the right-hand side

x^2 + 3x - 24 = 0

Step 3: Divide the coefficient of the x-term by 2

The coefficient of the x-term is 3. Divide 3 by 2 and square the result:

(3/2)^2 = 9/4

Step 4: Add the squared result to both sides

x^2 + 3x + 9/4 = 24 + 9/4

Step 5: Factor the left-hand side

(x + 3/2)^2 = 24 + 9/4

Step 6: Solve for x

Take the square root of both sides of the equation:

x + 3/2 = ±√(24 + 9/4)

x + 3/2 = ±√(97/4)

x + 3/2 = ±√97/2

x = -3/2 ± √97/2

x ≈ -3.55, 5.12

Example 2: Solving x^2 - 4x = 12

Let's use the equation x^2 - 4x = 12 as another example. To solve this equation by completing the square, follow the steps outlined above.

Step 1: Write the equation in the form ax^2 + bx + c = 0

x^2 - 4x = 12

Step 2: Move the constant term to the right-hand side

x^2 - 4x - 12 = 0

Step 3: Divide the coefficient of the x-term by 2

The coefficient of the x-term is -4. Divide -4 by 2 and square the result:

(-4/2)^2 = 16/4

Step 4: Add the squared result to both sides

x^2 - 4x + 16/4 = 12 + 16/4

Step 5: Factor the left-hand side

(x - 2)^2 = 12 + 16/4

Step 6: Solve for x

Take the square root of both sides of the equation:

x - 2 = ±√(12 + 16/4)

x - 2 = ±√(48/4)

x - 2 = ±√12

x - 2 = ±2√3

x = 2 ± 2√3

x ≈ 6.62, -0.62

Conclusion

Solving quadratic equations by completing the square is a powerful technique that can be used to solve equations of the form ax^2 + bx + c = 0. By following the steps outlined above, you can easily solve quadratic equations and find the solutions. Remember to round your answers to the nearest hundredth if necessary.

Practice Problems

Try solving the following quadratic equations by completing the square:

  1. x^2 + 5x = 30
  2. x^2 - 2x = 15
  3. x^2 + 2x = 20

Answer Key

  1. x ≈ 3.55, -6.55
  2. x ≈ 6.62, -0.62
  3. x ≈ 4.66, 5.12
    Frequently Asked Questions (FAQs) About Solving Quadratic Equations by Completing the Square =====================================================================================

Q: What is completing the square?

A: Completing the square is a technique used to solve quadratic equations of the form ax^2 + bx + c = 0. The method involves manipulating the equation to express it in the form of a perfect square trinomial, which can then be easily solved.

Q: What are the steps to complete the square?

A: The steps to complete the square are:

  1. Write the equation in the form ax^2 + bx + c = 0
  2. Move the constant term to the right-hand side
  3. Divide the coefficient of the x-term by 2
  4. Add the squared result to both sides
  5. Factor the left-hand side
  6. Solve for x

Q: How do I know if an equation can be solved by completing the square?

A: An equation can be solved by completing the square if it is in the form ax^2 + bx + c = 0, where a, b, and c are constants. If the equation is not in this form, it may not be possible to solve it by completing the square.

Q: What is a perfect square trinomial?

A: A perfect square trinomial is an expression of the form (x + d)^2, where d is a constant. Perfect square trinomials can be easily factored and solved.

Q: How do I factor a perfect square trinomial?

A: To factor a perfect square trinomial, follow these steps:

  1. Identify the binomial inside the parentheses
  2. Square the binomial
  3. Write the result as a perfect square trinomial

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

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

  • Not moving the constant term to the right-hand side
  • Not dividing the coefficient of the x-term by 2
  • Not adding the squared result to both sides
  • Not factoring the left-hand side
  • Not solving for x

Q: Can I use completing the square to solve all types of quadratic equations?

A: No, completing the square can only be used to solve quadratic equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. If the equation is not in this form, it may not be possible to solve it by completing the square.

Q: How do I know if I have completed the square correctly?

A: To check if you have completed the square correctly, follow these steps:

  1. Verify that the equation is in the form ax^2 + bx + c = 0
  2. Check that the constant term has been moved to the right-hand side
  3. Verify that the coefficient of the x-term has been divided by 2
  4. Check that the squared result has been added to both sides
  5. Verify that the left-hand side has been factored
  6. Check that the equation has been solved for x

Q: What are some real-world applications of completing the square?

A: Completing the square has many real-world applications, including:

  • Solving quadratic equations in physics and engineering
  • Modeling population growth and decline
  • Analyzing data and making predictions
  • Solving optimization problems

Conclusion

Completing the square is a powerful technique for solving quadratic equations. By following the steps outlined above and avoiding common mistakes, you can easily solve quadratic equations and find the solutions. Remember to round your answers to the nearest hundredth if necessary.

Practice Problems

Try solving the following quadratic equations by completing the square:

  1. x^2 + 5x = 30
  2. x^2 - 2x = 15
  3. x^2 + 2x = 20

Answer Key

  1. x ≈ 3.55, -6.55
  2. x ≈ 6.62, -0.62
  3. x ≈ 4.66, 5.12