Factoring by Completing the Square

Oct 13, 2025 by leolavanderia.com.br 0 views

Introduction

In mathematics, factoring quadratic expressions is a crucial skill that helps us solve equations, find roots, and understand the behavior of functions. The given expression $4x^2 - 4x - 120$ is a quadratic expression that can be factored in various ways. In this article, we will explore the different factored forms of this expression and determine which of the given options are equivalent to the original expression.

Factoring the Expression

To factor the expression $4x^2 - 4x - 120$ , we can start by looking for two numbers whose product is $-120$ and whose sum is $-4$ . These numbers are $-15$ and $8$ , since $-15 \times 8 = -120$ and $-15 + 8 = -7$ . However, we need to find two numbers whose sum is $-4$ , so we can rewrite the expression as $4x^2 - 4x - 120 = 4(x^2 - x) - 120$ .

Factoring by Grouping

We can factor the expression $4(x^2 - x) - 120$ by grouping the terms. We can rewrite the expression as $4(x^2 - x) - 120 = 4(x^2 - x - 10) + 80$ . Now, we can factor the quadratic expression inside the parentheses: $x^2 - x - 10 = (x - 5)(x + 2)$ . Therefore, we have factored the original expression as $4(x - 5)(x + 2) - 120$ .

Factoring by Completing the Square

Another way to factor the expression $4x^2 - 4x - 120$ is by completing the square. We can rewrite the expression as $4x^2 - 4x - 120 = 4(x^2 - x) - 120 = 4(x^2 - x + \frac{1}{4}) - 4(\frac{1}{4}) - 120 = 4(x - \frac{1}{2})^2 - 121$ .

Factoring by Difference of Squares

We can also factor the expression $4x^2 - 4x - 120$ by difference of squares. We can rewrite the expression as $4x^2 - 4x - 120 = (2x - 10)(2x + 12)$ .

Equivalent Forms

Now that we have factored the expression $4x^2 - 4x - 120$ in different ways, we can determine which of the given options are equivalent to the original expression. The options are:

I: $4(x - 5)(x + 2) - 120$
II: $4(x - \frac{1}{2})^2 - 121$
III: $(2x - 10)(2x + 12)$
IV: $4(x - 5)(x + 2)$

Conclusion

Based on our analysis, we can see that options II, III, and IV are equivalent to the original expression $4x^2 - 4x - 120$ . Option I is not equivalent to the original expression, since it has an additional term $-120$ .

Final Answer

The final answer is C. II, III, and IV.

Introduction

In our previous article, we explored the different factored forms of the expression $4x^2 - 4x - 120$ . We also determined which of the given options are equivalent to the original expression. In this article, we will answer some frequently asked questions related to the expression $4x^2 - 4x - 120$ .

Q&A

Q1: What is the factored form of the expression $4x^2 - 4x - 120$ ?

A1: The factored form of the expression $4x^2 - 4x - 120$ is $4(x - 5)(x + 2)$ .

Q2: How do you factor the expression $4x^2 - 4x - 120$ by completing the square?

A2: To factor the expression $4x^2 - 4x - 120$ by completing the square, we can rewrite the expression as $4(x^2 - x + \frac{1}{4}) - 4(\frac{1}{4}) - 120 = 4(x - \frac{1}{2})^2 - 121$ .

Q3: What is the difference of squares form of the expression $4x^2 - 4x - 120$ ?

A3: The difference of squares form of the expression $4x^2 - 4x - 120$ is $(2x - 10)(2x + 12)$ .

Q4: Which of the given options are equivalent to the original expression $4x^2 - 4x - 120$ ?

A4: Options II, III, and IV are equivalent to the original expression $4x^2 - 4x - 120$ .

Q5: How do you determine the equivalent forms of a quadratic expression?

A5: To determine the equivalent forms of a quadratic expression, we can factor the expression in different ways, such as by grouping, completing the square, or difference of squares.

Q6: What is the importance of factoring quadratic expressions?

A6: Factoring quadratic expressions is important because it helps us solve equations, find roots, and understand the behavior of functions.

Q7: Can you provide an example of a quadratic expression that can be factored in different ways?

A7: Yes, the expression $x^2 + 5x + 6$ can be factored in different ways, such as $(x + 3)(x + 2)$ or $(x + 2)(x + 3)$ .

Q8: How do you determine the number of solutions of a quadratic equation?

A8: To determine the number of solutions of a quadratic equation, we can use the discriminant, which is the expression under the square root in the quadratic formula.

Q9: What is the quadratic formula?

A9: The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation.

Q10: Can you provide an example of a quadratic equation that has two solutions?

A10: Yes, the equation $x^2 + 4x + 4 = 0$ has two solutions, which are $x = -2$ and $x = -2$ .

Conclusion

In this article, we answered some frequently asked questions related to the expression $4x^2 - 4x - 120$ . We also provided examples and explanations to help readers understand the concepts. We hope that this article has been helpful in clarifying any doubts that readers may have had.

Final Answer

The final answer is that the expression $4x^2 - 4x - 120$ can be factored in different ways, and options II, III, and IV are equivalent to the original expression.