The Product Of -7 And A Number Squared Is -28

The Product of -7 and a Number Squared is -28: A Comprehensive Exploration

This article delves into the algebraic equation "-7x² = -28," exploring its solution, variations, and broader mathematical concepts. We'll examine different approaches to solving this quadratic equation, highlighting the importance of understanding fundamental algebraic principles. Furthermore, we'll discuss practical applications and extensions of this seemingly simple problem.

Understanding the Problem: -7x² = -28

The core of the problem lies in finding the value(s) of 'x' that satisfy the equation -7x² = -28. This is a quadratic equation because the highest power of the variable 'x' is 2. Solving it involves isolating 'x' and finding its possible values. The equation represents a mathematical relationship where the product of -7 and the square of a number is equal to -28.

Deconstructing the Equation

Let's break down the equation step-by-step:

• -7x²: This term represents the product of -7 and the square of an unknown number (x²). The square of a number is always positive or zero (x² ≥ 0).
• = -28: This indicates that the result of the multiplication (-7x²) is equal to -28.

Our goal is to determine the value(s) of 'x' that make this statement true.

Solving the Equation: Step-by-Step Guide

There are several methods to solve this quadratic equation. We'll explore two common and effective approaches:

Method 1: Isolating x²

This is the most straightforward method. We begin by isolating the x² term:

1. Divide both sides by -7: This eliminates the coefficient of x². The equation becomes: x² = 4

2. Take the square root of both sides: This will give us the value of x. Remember that the square root of a number can be both positive and negative. Therefore: x = ±√4 x = ±2

Therefore, the solutions to the equation are x = 2 and x = -2.

Method 2: Factoring

While less efficient for this specific problem, factoring provides a valuable understanding of quadratic equations and is crucial for more complex scenarios. We can rewrite the equation as:

-7x² + 28 = 0

Now, we can factor out a -7:

-7(x² - 4) = 0

This expression can be further factored using the difference of squares:

-7(x - 2)(x + 2) = 0

This equation is satisfied if any of the factors are equal to zero. Therefore:

• x - 2 = 0 => x = 2
• x + 2 = 0 => x = -2

Again, we arrive at the solutions x = 2 and x = -2.

Verification of Solutions

It's crucial to verify our solutions by substituting them back into the original equation:

For x = 2:

-7(2)² = -7(4) = -28. This is correct.

For x = -2:

-7(-2)² = -7(4) = -28. This is also correct.

Graphical Representation

The equation -7x² = -28 can be represented graphically as a parabola. The parabola intersects the x-axis at x = 2 and x = -2, visually confirming our solutions. The parabola opens downwards due to the negative coefficient of the x² term.

Extending the Concept: Variations and Generalizations

The fundamental principles explored here can be applied to a broader range of quadratic equations. Let's consider some variations:

• -ax² = b: This represents a general form of the equation. To solve, divide both sides by -a, then take the square root. The solutions will be x = ±√(b/-a), provided b/-a is non-negative.

• -ax² + c = 0: This introduces a constant term. You can solve this using factoring, the quadratic formula, or completing the square.

• ax² + bx + c = 0: This is the standard form of a quadratic equation. The quadratic formula is generally the most efficient method for solving this type of equation.

Real-World Applications

While seemingly abstract, quadratic equations have numerous real-world applications:

• Physics: Calculating projectile motion, determining the trajectory of objects under gravity.

• Engineering: Designing structures, optimizing shapes and materials.

• Economics: Modeling supply and demand, predicting market trends.

• Computer Graphics: Creating curves and shapes for animation and game development.

Conclusion: Mastering Quadratic Equations

Solving the equation -7x² = -28 provides a foundational understanding of quadratic equations and their solutions. Mastering these concepts is crucial for further advancements in algebra, calculus, and various scientific and engineering disciplines. By understanding the different solution methods and their applications, we can effectively tackle more complex mathematical problems and appreciate the power of algebraic principles in solving real-world challenges. Remember to always verify your solutions and explore different approaches to deepen your understanding. The more you practice, the more confident and proficient you'll become in solving quadratic equations and other algebraic problems.