30 Increased By 3 Times The Square Of A Number

Decoding the Mystery: 30 Increased by 3 Times the Square of a Number

This article delves into the mathematical expression "30 increased by 3 times the square of a number," exploring its meaning, how to represent it algebraically, solving related equations, and expanding upon its applications in various mathematical contexts. Understanding this seemingly simple phrase opens doors to a wider appreciation of algebraic manipulation and problem-solving techniques. We'll explore this concept thoroughly, progressing from basic understanding to more complex scenarios, ensuring a clear and comprehensive understanding for all readers. This exploration will equip you with the skills to tackle similar problems and further your understanding of algebra.

Understanding the Phrase

The phrase "30 increased by 3 times the square of a number" describes a mathematical operation. Let's break it down step-by-step:

• A number: This represents an unknown value, often symbolized by a variable, typically 'x' or 'n'.
• The square of a number: This means the number multiplied by itself (n²).
• 3 times the square of a number: This is the result of multiplying the square of the number by 3 (3n²).
• 30 increased by 3 times the square of a number: This means adding 30 to the result of the previous step (30 + 3n²).

Algebraic Representation

The phrase can be concisely represented by a simple algebraic expression: 30 + 3n² where 'n' represents the unknown number. This expression forms the basis for many mathematical problems and equations.

Solving Equations: Finding the Unknown Number

The algebraic expression becomes an equation when we set it equal to a specific value. Let's explore different scenarios:

Scenario 1: The expression equals 48

If "30 increased by 3 times the square of a number" equals 48, we can formulate the following equation:

30 + 3n² = 48

To solve for 'n', we follow these steps:

1. Subtract 30 from both sides: 3n² = 18
2. Divide both sides by 3: n² = 6
3. Take the square root of both sides: n = ±√6

Therefore, there are two possible solutions for 'n': positive √6 and negative √6. This highlights the importance of considering both positive and negative roots when solving quadratic equations.

Scenario 2: The expression equals 0

If the expression equals 0, we have:

30 + 3n² = 0

Solving for 'n':

1. Subtract 30 from both sides: 3n² = -30
2. Divide both sides by 3: n² = -10

In this case, there are no real solutions for 'n'. The square of a real number cannot be negative. The solutions involve imaginary numbers, which are beyond the scope of this introductory explanation.

Scenario 3: The expression is greater than a certain value

Let's consider an inequality: 30 + 3n² > 60

Solving this inequality:

1. Subtract 30 from both sides: 3n² > 30
2. Divide both sides by 3: n² > 10
3. Take the square root of both sides (remembering to consider both positive and negative roots and the direction of the inequality): n > √10 or n < -√10

This means that the inequality holds true for any value of 'n' greater than √10 or less than -√10.

Graphical Representation

The expression 30 + 3n² represents a parabola, a U-shaped curve. Graphing this equation provides a visual representation of the relationship between 'n' and the value of the expression. The parabola opens upwards, indicating that the expression's value increases as the absolute value of 'n' increases. The y-intercept (where n=0) is 30.

Applications and Further Exploration

The expression "30 increased by 3 times the square of a number" isn't just a theoretical exercise; it has practical applications in various fields:

• Physics: Equations describing motion, energy, and other physical phenomena often involve quadratic expressions. For instance, the kinetic energy of an object is proportional to the square of its velocity.

• Engineering: Designing structures, calculating forces, and analyzing stress often require solving quadratic equations.

• Economics: Quadratic functions can model certain economic relationships, such as cost functions or revenue functions.

• Computer Science: Quadratic equations are frequently encountered in algorithms and data structures.

• Statistics: Quadratic regression analysis is used to model data that exhibits a quadratic relationship.

Expanding the Concept: Variations and Extensions

We can expand the concept by introducing variations:

• Changing the constant: Instead of 30, we could use any other constant value. This would shift the parabola vertically.

• Changing the coefficient of n²: Changing the '3' to another number would alter the steepness of the parabola. A larger coefficient would make the parabola narrower, and a smaller coefficient would make it wider.

• Adding linear terms: We could add a term involving 'n' (e.g., 30 + 3n² + 2n), resulting in a more complex quadratic expression.

These variations lead to more intricate equations and graphs, but the fundamental principles of algebraic manipulation and problem-solving remain the same.

Frequently Asked Questions (FAQ)

Q1: What if the number is negative?

A: The expression works perfectly well with negative numbers. Squaring a negative number results in a positive number, so the overall expression will still be positive (as long as the constant is large enough). Remember to account for the negative sign when solving equations.

Q2: Are there always two solutions when solving for 'n'?

A: Not always. As we saw in Scenario 2, there may be no real solutions if the resulting quadratic equation has no real roots. There could also be only one solution if the quadratic equation is a perfect square.

Q3: How can I solve more complex variations of this expression?

A: More complex variations often require more advanced algebraic techniques such as factoring, completing the square, or using the quadratic formula. These techniques allow you to solve for 'n' even in more challenging scenarios.

Q4: What are the limitations of this expression?

A: The expression itself is a simplification of reality. It might represent a good approximation in certain contexts, but it may not be accurate in all situations. The applicability depends on the specific problem you are trying to model.

Conclusion

The seemingly simple phrase "30 increased by 3 times the square of a number" reveals a rich mathematical landscape. By understanding its algebraic representation, solving related equations, and exploring its graphical representation, we gain valuable insights into the power of algebra. This understanding extends beyond simple equation solving; it forms the basis for tackling more complex mathematical problems across various disciplines. This exploration should serve as a stepping stone to further delve into the world of algebra and its applications. Remember to practice regularly to solidify your understanding and build confidence in your problem-solving abilities. The journey into mathematics is a rewarding one, filled with discovery and intellectual stimulation.