Solve Each Equation By Completing The Square.$\[ 12x^2 + 16x - 3 = 0 \\]

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Introduction

Completing the square is a powerful technique used to solve quadratic equations. It involves manipulating the equation to express it in a perfect square form, which can then be easily solved. In this article, we will focus on solving the equation 12x^2 + 16x - 3 = 0 by completing the square.

What is Completing the Square?

Completing the square is a method of solving quadratic equations by rewriting them in a perfect square form. This involves adding and subtracting a constant term to create a perfect square trinomial. The constant term is found by taking half of the coefficient of the x-term and squaring it.

Step 1: Move the Constant Term to the Right Side

The first step in completing the square is to move the constant term to the right side of the equation. This gives us:

12x^2 + 16x = 3

Step 2: Factor Out the Coefficient of x^2

Next, we need to factor out the coefficient of x^2 from the left side of the equation. This gives us:

12(x^2 + (16/12)x) = 3

Step 3: Simplify the Coefficient of x

We can simplify the coefficient of x by dividing 16 by 12. This gives us:

12(x^2 + (4/3)x) = 3

Step 4: Find the Constant Term to Add

To complete the square, we need to add a constant term to the left side of the equation. This constant term is found by taking half of the coefficient of x and squaring it. In this case, the coefficient of x is 4/3, so we need to add (4/3)^2 = 16/9 to the left side of the equation.

Step 5: Add the Constant Term and Simplify

We can now add the constant term to the left side of the equation and simplify:

12(x^2 + (4/3)x + 16/36) = 3 + 12(16/36)

Step 6: Factor the Perfect Square Trinomial

The left side of the equation is now a perfect square trinomial, which can be factored as:

12(x + 4/6)^2 = 3 + 12(16/36)

Step 7: Simplify the Right Side

We can simplify the right side of the equation by combining the terms:

12(x + 4/6)^2 = 3 + 4

Step 8: Solve for x

We can now solve for x by taking the square root of both sides of the equation:

x + 4/6 = ±√(7/12)

Step 9: Simplify the Square Root

We can simplify the square root by rationalizing the denominator:

x + 4/6 = ±√(7/12) × √(12/12)

Step 10: Simplify the Expression

We can simplify the expression by combining the terms:

x + 4/6 = ±√(84/144)

Step 11: Simplify the Fraction

We can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor:

x + 4/6 = ±√(7/16)

Step 12: Solve for x

We can now solve for x by isolating the variable:

x = -4/6 ± √(7/16)

Step 13: Simplify the Expression

We can simplify the expression by combining the terms:

x = -2/3 ± √(7/16)

Conclusion

In this article, we have solved the equation 12x^2 + 16x - 3 = 0 by completing the square. We have followed the steps outlined above to rewrite the equation in a perfect square form and then solved for x. The final solution is x = -2/3 ± √(7/16).

Example Problems

Here are a few example problems that you can try to practice completing the square:

  • 2x^2 + 6x - 1 = 0
  • 3x^2 - 2x - 5 = 0
  • x^2 + 4x + 4 = 0

Tips and Tricks

Here are a few tips and tricks to help you complete the square:

  • Make sure to move the constant term to the right side of the equation before starting the process.
  • Factor out the coefficient of x^2 from the left side of the equation.
  • Simplify the coefficient of x by dividing the numerator and denominator by their greatest common divisor.
  • Add the constant term to the left side of the equation and simplify.
  • Factor the perfect square trinomial and simplify the right side of the equation.
  • Solve for x by taking the square root of both sides of the equation.

Common Mistakes

Here are a few common mistakes to avoid when completing the square:

  • Failing to move the constant term to the right side of the equation.
  • Failing to factor out the coefficient of x^2 from the left side of the equation.
  • Failing to simplify the coefficient of x by dividing the numerator and denominator by their greatest common divisor.
  • Failing to add the constant term to the left side of the equation and simplify.
  • Failing to factor the perfect square trinomial and simplify the right side of the equation.
  • Failing to solve for x by taking the square root of both sides of the equation.

Conclusion

Q: What is completing the square?

A: Completing the square is a method of solving quadratic equations by rewriting them in a perfect square form. This involves adding and subtracting a constant term to create a perfect square trinomial.

Q: Why is completing the square useful?

A: Completing the square is useful because it allows us to solve quadratic equations that cannot be factored easily. It also helps us to find the solutions to quadratic equations in a more efficient and accurate way.

Q: What are the steps involved in completing the square?

A: The steps involved in completing the square are:

  1. Move the constant term to the right side of the equation.
  2. Factor out the coefficient of x^2 from the left side of the equation.
  3. Simplify the coefficient of x by dividing the numerator and denominator by their greatest common divisor.
  4. Add the constant term to the left side of the equation and simplify.
  5. Factor the perfect square trinomial and simplify the right side of the equation.
  6. Solve for x by taking the square root of both sides of the equation.

Q: What is the formula for completing the square?

A: The formula for completing the square is:

x^2 + bx + c = (x + b/2)^2 - (b/2)^2 + c

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

A: You should use completing the square when the quadratic equation cannot be factored easily, or when you need to find the solutions to the equation in a more efficient and accurate way.

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 move the constant term to the right side of the equation.
  • Failing to factor out the coefficient of x^2 from the left side of the equation.
  • Failing to simplify the coefficient of x by dividing the numerator and denominator by their greatest common divisor.
  • Failing to add the constant term to the left side of the equation and simplify.
  • Failing to factor the perfect square trinomial and simplify the right side of the equation.
  • Failing to solve for x by taking the square root of both sides of the equation.

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 will need to use complex numbers and follow the same steps as before.

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 will need to follow the same steps as before and simplify the rational expressions.

Q: How do I check my work when completing the square?

A: To check your work when completing the square, you should:

  • Plug the solutions back into the original equation to make sure they are true.
  • Simplify the equation to make sure it is in the correct form.
  • Check that the solutions are in the correct form (e.g. x = ±√(expression)).

Q: Can I use completing the square to solve quadratic equations with multiple variables?

A: No, you cannot use completing the square to solve quadratic equations with multiple variables. Completing the square is only used to solve quadratic equations with one variable.

Q: Can I use completing the square to solve quadratic equations with non-integer coefficients?

A: Yes, you can use completing the square to solve quadratic equations with non-integer coefficients. However, you will need to follow the same steps as before and simplify the non-integer expressions.

Conclusion

Completing the square is a powerful technique used to solve quadratic equations. By following the steps outlined above and avoiding common mistakes, you can use completing the square to solve quadratic equations with complex coefficients, rational coefficients, and non-integer coefficients. Remember to check your work and simplify the expressions to ensure that you have found the correct solutions.