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

The Product of -7 and a Number Squared is -28: Solving the Equation and Exploring Related Concepts

This article delves into the mathematical problem: "The product of -7 and a number squared is -28." We'll not only solve this specific equation but also explore the broader mathematical concepts involved, including quadratic equations, the concept of square roots, and the importance of understanding both positive and negative solutions. We'll also touch upon real-world applications and extend the problem to more complex scenarios.

Understanding the Problem: Breaking Down the Equation

The problem statement, "The product of -7 and a number squared is -28," can be translated into a mathematical equation. Let's represent the unknown number with the variable 'x'. The equation becomes:

-7x² = -28

This is a simple quadratic equation. Solving it will reveal the value(s) of 'x' that satisfy the equation.

Solving the Quadratic Equation: Step-by-Step Guide

To solve for 'x', we'll follow these steps:

1. Isolate the x² term:

Divide both sides of the equation by -7:

x² = (-28) / (-7)

x² = 4

2. Find the square root:

To solve for x, we need to take the square root of both sides:

√x² = ±√4

This step introduces the crucial concept of both positive and negative solutions. Remember, both +2 and -2, when squared, result in 4.

3. Determine the Solutions:

Therefore, the solutions to the equation are:

x = +2 and x = -2

Verification: Checking Our Solutions

It's always a good practice to check our solutions by substituting them back into the original equation:

For x = 2:

-7(2)² = -7(4) = -28 (Correct!)

For x = -2:

-7(-2)² = -7(4) = -28 (Correct!)

Both solutions are valid.

Expanding the Understanding: Exploring Quadratic Equations

Our initial problem was a specific instance of a quadratic equation. The general form of a quadratic equation is:

ax² + bx + c = 0

where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero.

Our problem, -7x² = -28, can be rewritten in the standard quadratic form by adding 28 to both sides:

-7x² + 28 = 0

Here, a = -7, b = 0, and c = 28. The fact that 'b' is zero simplifies the equation, making it easier to solve using the square root method. More complex quadratic equations with non-zero 'b' values may require the quadratic formula or factoring techniques to solve.

The Importance of Positive and Negative Solutions

Understanding that quadratic equations often have two solutions is crucial. In many real-world applications, both positive and negative solutions can hold significant meaning, depending on the context of the problem. In some cases, only one solution might be physically possible or relevant (e.g., a negative length is impossible). In others, both solutions are equally valid.

Real-World Applications: Where Quadratic Equations are Used

Quadratic equations are not just abstract mathematical concepts; they have numerous applications in various fields:

• Physics: Calculating projectile motion, determining the trajectory of objects under the influence of gravity.
• Engineering: Designing structures, bridges, and other engineering marvels, where understanding stress and strain is critical.
• Finance: Modeling financial growth or decay over time, predicting future values based on initial investments or debts.
• Computer Graphics: Creating curves and shapes in computer-generated imagery (CGI).

Extending the Problem: More Challenging Scenarios

Let's consider some variations of the original problem to further strengthen our understanding:

Scenario 1: A slightly more complex equation

Suppose the problem was: "The product of -7 and a number squared, plus 14, is 0." This translates to the equation:

-7x² + 14 = 0

This equation can be solved using similar steps as before:

1. Subtract 14 from both sides: -7x² = -14
2. Divide by -7: x² = 2
3. Take the square root: x = ±√2

In this case, the solutions are irrational numbers (involving the square root of 2).

Scenario 2: Introducing a linear term

Let's add a linear term to the equation. For example: "The product of -7 and a number squared, plus 5 times the number, is -28." This equation becomes:

-7x² + 5x + 28 = 0

Solving this equation requires the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

Where a = -7, b = 5, and c = 28. Substituting these values into the quadratic formula will yield the two solutions for 'x'.

Conclusion: Mastering Quadratic Equations

Solving the equation "-7x² = -28" is a foundational step in understanding quadratic equations. By working through the problem and exploring related concepts, we've not only found the solutions (x = ±2) but also gained a deeper appreciation for the broader mathematical principles involved. Understanding quadratic equations is crucial for various applications in numerous fields. Mastering the techniques for solving these equations, including using the quadratic formula, will empower you to tackle more complex mathematical problems and apply these principles to solve real-world challenges. The ability to interpret and solve quadratic equations is a valuable skill that extends far beyond the classroom.