What Number Must You Add To Complete The Square For The Equation $x^2 + 4x = 15$?A. 4 B. 16 C. 2 D. 8

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 focus on completing the square for the equation $x^2 + 4x = 15$ and determine the number that must be added to complete the square.

Understanding the Concept of Completing the Square

Completing the square is a method of solving quadratic equations by expressing them in a perfect square trinomial form. This involves adding and subtracting a constant term to create a perfect square trinomial. The constant term is determined by the coefficient of the linear term in the quadratic equation.

The Formula for Completing the Square

The formula for completing the square is:

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

where $b$ is the coefficient of the linear term and $c$ is the constant term.

Applying the Formula to the Given Equation

For the equation $x^2 + 4x = 15$, we can apply the formula for completing the square by substituting $b = 4$ and $c = 15$.

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

Simplifying the Equation

Simplifying the equation, we get:

$$x^2 + 4x + 4 = 15 + 4$$

$$x^2 + 4x + 4 = 19$$

Expressing the Equation in Perfect Square Trinomial Form

The equation can now be expressed in perfect square trinomial form as:

$$(x + 2)^2 = 19$$

Determining the Number to Add to Complete the Square

To complete the square, we need to add the constant term $\left(\frac{b}{2}\right)^2$ to both sides of the equation. In this case, the constant term is $\left(\frac{4}{2}\right)^2 = 4$. Therefore, the number that must be added to complete the square is 4.

Conclusion

In conclusion, the number that must be added to complete the square for the equation $x^2 + 4x = 15$ is 4. This is determined by applying the formula for completing the square and simplifying the equation to express it in perfect square trinomial form.

Frequently Asked Questions

• What is completing the square? Completing the square is a method of solving quadratic equations by expressing them in a perfect square trinomial form.
• How do I apply the formula for completing the square? To apply the formula, substitute the coefficient of the linear term and the constant term into the formula and simplify the equation.
• What is the number that must be added to complete the square? The number that must be added to complete the square is determined by the formula for completing the square and is equal to $\left(\frac{b}{2}\right)^2$.

Final Answer

The final answer is: $\boxed{4}$

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 provide a Q&A guide to completing the square, covering common questions and topics related to this technique.

Q&A Guide

Q: What is completing the square?

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

Q: How do I apply the formula for completing the square?

A: To apply the formula, substitute the coefficient of the linear term and the constant term into the formula and simplify the equation. The formula for completing the square is:

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

Q: What is the number that must be added to complete the square?

A: The number that must be added to complete the square is determined by the formula for completing the square and is equal to $\left(\frac{b}{2}\right)^2$.

Q: How do I determine the value of b in the formula?

A: The value of b is the coefficient of the linear term in the quadratic equation. For example, in the equation $x^2 + 4x = 15$, the value of b is 4.

Q: Can I use completing the square to solve any quadratic equation?

A: Yes, completing the square can be used to solve any quadratic equation. However, it is most useful for equations that are not easily factored or solved using other methods.

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

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

• Not adding and subtracting the constant term correctly
• Not simplifying the equation correctly
• Not checking the solution to ensure it is valid

Q: How do I check the solution to ensure it is valid?

A: To check the solution, substitute the value of x back into the original equation and simplify. If the equation is true, then the solution is valid.

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

A: Yes, completing the square can be used to solve quadratic equations with complex coefficients. However, the process is slightly more complicated and requires the use of complex numbers.

Q: How do I apply completing the square to solve quadratic equations with complex coefficients?

A: To apply completing the square to solve quadratic equations with complex coefficients, substitute the complex coefficients into the formula and simplify the equation. The process is similar to solving quadratic equations with real coefficients, but requires the use of complex numbers.

Conclusion

In conclusion, completing the square is a powerful technique used to solve quadratic equations. By understanding the formula and process for completing the square, you can solve a wide range of quadratic equations. Remember to check the solution to ensure it is valid and to avoid common mistakes when completing the square.

Frequently Asked Questions

• What is completing the square? Completing the square is a method of solving quadratic equations by expressing them in a perfect square trinomial form.
• How do I apply the formula for completing the square? To apply the formula, substitute the coefficient of the linear term and the constant term into the formula and simplify the equation.
• What is the number that must be added to complete the square? The number that must be added to complete the square is determined by the formula for completing the square and is equal to $\left(\frac{b}{2}\right)^2$.

Final Answer

The final answer is: $\boxed{4}$