Identifying the Quadratic Expression

Oct 13, 2025 by leolavanderia.com.br 41 views

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Introduction

In mathematics, solving equations involving consecutive integers can be a fascinating and challenging task. In this article, we will delve into the world of quadratic equations and explore how to find the constant term in a quadratic expression when given the product of two consecutive integers. We will use the example of the product of two consecutive integers being 420 to derive the equation and solve for the smaller integer.

The Problem

The problem states that the product of two consecutive integers is 420. Let's denote the smaller integer as x. Since the integers are consecutive, the next integer can be represented as x + 1. Therefore, we can write the equation as:

x(x + 1) = 420

Expanding the Equation

To expand the equation, we multiply the two binomials:

x^2 + x = 420

Rearranging the Equation

To set the equation to 0, we subtract 420 from both sides:

x^2 + x - 420 = 0

Identifying the Quadratic Expression

The equation x^2 + x - 420 = 0 is a quadratic expression in standard form. The constant term in this expression is -420.

Factoring the Quadratic Expression

To factor the quadratic expression, we need to find two numbers whose product is -420 and whose sum is 1. These numbers are 21 and -20, since 21 * (-20) = -420 and 21 + (-20) = 1.

Writing the Factored Form

Using the numbers 21 and -20, we can write the factored form of the quadratic expression as:

(x + 21)(x - 20) = 0

Solving for x

To solve for x, we set each factor equal to 0 and solve for x:

x + 21 = 0 --> x = -21 x - 20 = 0 --> x = 20

Conclusion

In this article, we explored how to find the constant term in a quadratic expression when given the product of two consecutive integers. We used the example of the product of two consecutive integers being 420 to derive the equation and solve for the smaller integer. The constant term in the quadratic expression x^2 + x - 420 = 0 is -420.

The Constant of the Quadratic Expression

The constant of the quadratic expression in the equation x^2 + x - 420 = 0 is -420.

Final Answer

The final answer is -420.

Frequently Asked Questions

Q: What is the product of two consecutive integers?

A: The product of two consecutive integers is 420.

Q: What is the equation in standard form to solve for the smaller integer?

A: The equation in standard form is x^2 + x - 420 = 0.

Q: What is the constant term in the quadratic expression?

A: The constant term in the quadratic expression is -420.

Q: How do you factor the quadratic expression?

A: To factor the quadratic expression, you need to find two numbers whose product is -420 and whose sum is 1.

Q: What are the solutions to the equation?

A: The solutions to the equation are x = -21 and x = 20.

References

[1] "Quadratic Equations" by Math Open Reference
[2] "Factoring Quadratic Expressions" by Purplemath

The Product of Two Consecutive Integers: A Mathematical Exploration
Solving Quadratic Equations by Factoring
The Constant of the Quadratic Expression