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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 trinomial form, which can be easily factored or solved. In this article, we will explore the concept of completing the square and provide a step-by-step guide on how to solve quadratic equations using this method.

What is Completing the Square?

Completing the square is a process of transforming a quadratic equation into a perfect square trinomial form. This is done by adding and subtracting a constant term to the equation, which allows us to express it in a form that can be easily factored or solved. The constant term added is called the "square root of the coefficient of the x-term divided by 2".

The Formula for Completing the Square

The formula for completing the square is:

$$x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$$

where b is the coefficient of the x-term.

Step-by-Step Guide to Completing the Square

To complete the square, follow these steps:

1. Write the quadratic equation in the form $x^2 + bx + c = 0$.
2. Identify the coefficient of the x-term (b).
3. Calculate the constant term to be added ( $\left(\frac{b}{2}\right)^2$ ).
4. Add the constant term to both sides of the equation.
5. Express the left-hand side of the equation as a perfect square trinomial.
6. Solve for x by taking the square root of both sides of the equation.

Example: Completing the Square

Let's consider the quadratic equation:

$$x^2 + 15x = 49$$

To complete the square, we need to add a constant term to both sides of the equation. The coefficient of the x-term is 15, so we need to calculate the constant term to be added:

$$\left(\frac{15}{2}\right)^2 = \left(\frac{15}{2}\right) \times \left(\frac{15}{2}\right) = \frac{225}{4}$$

Now, we add the constant term to both sides of the equation:

$$x^2 + 15x + \frac{225}{4} = 49 + \frac{225}{4}$$

Simplifying the right-hand side of the equation, we get:

$$x^2 + 15x + \frac{225}{4} = \frac{196 + 225}{4}$$

$$x^2 + 15x + \frac{225}{4} = \frac{421}{4}$$

Now, we express the left-hand side of the equation as a perfect square trinomial:

$$\left(x + \frac{15}{2}\right)^2 = \frac{421}{4}$$

Taking the square root of both sides of the equation, we get:

$$x + \frac{15}{2} = \pm \sqrt{\frac{421}{4}}$$

Simplifying the right-hand side of the equation, we get:

$$x + \frac{15}{2} = \pm \frac{\sqrt{421}}{2}$$

Solving for x, we get:

$$x = -\frac{15}{2} \pm \frac{\sqrt{421}}{2}$$

Conclusion

Completing the square is a powerful technique used to solve quadratic equations. By following the steps outlined in this article, you can easily complete the square and solve quadratic equations. Remember to identify the coefficient of the x-term, calculate the constant term to be added, add the constant term to both sides of the equation, express the left-hand side of the equation as a perfect square trinomial, and solve for x by taking the square root of both sides of the equation.

Frequently Asked Questions

• What is completing the square? Completing the square is a process of transforming a quadratic equation into a perfect square trinomial form.
• How do I complete the square? To complete the square, follow the steps outlined in this article.
• What is the formula for completing the square? The formula for completing the square is: $x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$
• How do I solve a quadratic equation using completing the square? To solve a quadratic equation using completing the square, follow the steps outlined in this article.

References

• "Algebra" by Michael Artin
• "Calculus" by Michael Spivak
• "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

• "Completing the Square: A Step-by-Step Guide" by Math Open Reference
• "Completing the Square: A Tutorial" by Purplemath
• "Completing the Square: A Guide" by Khan Academy
  

Introduction

Completing the square is a powerful technique used to solve quadratic equations. However, it can be a bit confusing, especially for those who are new to algebra. In this article, we will answer some of the most frequently asked questions about completing the square.

Q: What is completing the square?

A: Completing the square is a process of transforming a quadratic equation into a perfect square trinomial form. This is done by adding and subtracting a constant term to the equation, which allows us to express it in a form that can be easily factored or solved.

Q: How do I complete the square?

A: To complete the square, follow these steps:

1. Write the quadratic equation in the form $x^2 + bx + c = 0$.
2. Identify the coefficient of the x-term (b).
3. Calculate the constant term to be added ( $\left(\frac{b}{2}\right)^2$ ).
4. Add the constant term to both sides of the equation.
5. Express the left-hand side of the equation as a perfect square trinomial.
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:

$$x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$$

Q: How do I solve a quadratic equation using completing the square?

A: To solve a quadratic equation using completing the square, follow the steps outlined above.

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

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

• Not identifying the coefficient of the x-term correctly.
• Not calculating the constant term to be added correctly.
• Not adding the constant term to both sides of the equation.
• Not expressing the left-hand side of the equation as a perfect square trinomial.

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

A: No, completing the square is not suitable for all types of quadratic equations. It is best used for quadratic equations that can be written in the form $x^2 + bx + c = 0$.

Q: How do I know if I should use completing the square or another method to solve a quadratic equation?

A: To determine whether to use completing the square or another method to solve a quadratic equation, try the following:

• Check if the quadratic equation can be written in the form $x^2 + bx + c = 0$.
• Check if the coefficient of the x-term is a rational number.
• Check if the constant term is a rational number.

If the quadratic equation meets these conditions, completing the square may be a suitable method to use.

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

A: No, completing the square is not suitable for quadratic equations with complex coefficients.

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

A: No, completing the square is not suitable for quadratic equations with irrational coefficients.

Conclusion

Completing the square is a powerful technique used to solve quadratic equations. However, it can be a bit confusing, especially for those who are new to algebra. By following the steps outlined in this article, you can easily complete the square and solve quadratic equations.

Frequently Asked Questions

• What is completing the square? Completing the square is a process of transforming a quadratic equation into a perfect square trinomial form.
• How do I complete the square? To complete the square, follow the steps outlined in this article.
• What is the formula for completing the square? The formula for completing the square is: $x^2 + bx + \left(\frac{b}{2}\right)^2 = \left(x + \frac{b}{2}\right)^2$
• How do I solve a quadratic equation using completing the square? To solve a quadratic equation using completing the square, follow the steps outlined in this article.

References

• "Algebra" by Michael Artin
• "Calculus" by Michael Spivak
• "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

• "Completing the Square: A Step-by-Step Guide" by Math Open Reference
• "Completing the Square: A Tutorial" by Purplemath
• "Completing the Square: A Guide" by Khan Academy