30 Increased By 3 Times The Square Of A Number

30 Increased by 3 Times the Square of a Number: A Comprehensive Exploration

This article delves into the mathematical expression "30 increased by 3 times the square of a number," exploring its various interpretations, solutions, and applications. We'll dissect the problem, examine different approaches to solving it, and discuss its relevance in various mathematical contexts. The goal is to provide a comprehensive understanding, suitable for students and enthusiasts alike.

Understanding the Problem Statement

The phrase "30 increased by 3 times the square of a number" translates directly into a mathematical equation. Let's break it down:

• "a number": This represents an unknown value, typically denoted by a variable, such as x.
• "the square of a number": This means the number multiplied by itself, or x².
• "3 times the square of a number": This translates to 3 * x² or 3x².
• "30 increased by...": This signifies adding 30 to the previous expression.

Therefore, the complete mathematical representation is: 30 + 3x²

This expression forms the basis of various mathematical problems, from simple equation solving to more complex applications involving quadratic functions and their graphs.

Solving the Equation: Finding the Value of 'x'

The problem's difficulty depends on the context. Are we simply asked to write the expression, or are we given a specific value for the entire expression and asked to solve for x? Let's explore both scenarios:

Scenario 1: Expressing the Relationship

If the problem simply requires expressing "30 increased by 3 times the square of a number" mathematically, the answer is straightforward: 30 + 3x². This represents the relationship between the number (x) and the resulting value.

Scenario 2: Solving for 'x' Given a Value

This scenario involves setting the expression equal to a specific value and solving the resulting quadratic equation. For example:

Problem: "30 increased by 3 times the square of a number is equal to 57. Find the number."

Equation: 30 + 3x² = 57

Solution:

1. Isolate the x² term: Subtract 30 from both sides: 3x² = 27

2. Solve for x²: Divide both sides by 3: x² = 9

3. Solve for x: Take the square root of both sides: x = ±3

Therefore, the number could be either 3 or -3. Both values satisfy the original equation. This demonstrates that quadratic equations often have multiple solutions.

Graphical Representation: Visualizing the Quadratic Function

The expression 30 + 3x² represents a quadratic function. Quadratic functions are characterized by their parabolic graphs. Let's examine the graphical representation of this function:

• Parabola: The graph of 30 + 3x² is a parabola that opens upwards (since the coefficient of x² is positive).
• Vertex: The vertex represents the minimum value of the function. In this case, the vertex is at (0, 30). This means the minimum value of the expression is 30, which occurs when x = 0.
• Symmetry: Parabolas are symmetrical. The line of symmetry for this parabola is the y-axis (x = 0).
• x-intercepts: These are the points where the parabola intersects the x-axis (where the y-value is 0). Finding these points involves solving the equation 30 + 3x² = 0, which has no real solutions because the discriminant is negative. This indicates that the parabola lies entirely above the x-axis.

Understanding the graph provides a visual representation of the relationship between x and the expression's value. This visual aid can be invaluable in interpreting the results and understanding the function's behavior.

Applications and Extensions

The concept of "30 increased by 3 times the square of a number" has applications in various mathematical fields and real-world scenarios:

• Physics: Quadratic equations are frequently used in physics to model projectile motion, where the height of an object is a function of time.
• Engineering: Designing structures and calculating optimal dimensions often involves quadratic equations.
• Finance: Compound interest calculations involve quadratic and higher-order equations.
• Optimization Problems: Many optimization problems in various fields can be modeled and solved using quadratic equations to find maximum or minimum values.
• Computer Graphics: Creating curved shapes and paths in computer graphics often employs quadratic functions and their properties.

Advanced Concepts and Related Topics

To further enhance your understanding, let's touch upon some related mathematical concepts:

• Completing the Square: This technique is used to solve quadratic equations by rewriting the equation in a form that allows for easy extraction of the roots.
• Quadratic Formula: The quadratic formula provides a general solution for any quadratic equation of the form ax² + bx + c = 0. It's a powerful tool for solving equations that are difficult to factor.
• Discriminant: The discriminant (b² - 4ac) of a quadratic equation determines the nature of the roots (real and distinct, real and equal, or complex).
• Parabola Properties: Understanding the properties of parabolas, such as their focus, directrix, and axis of symmetry, provides deeper insights into the behavior of quadratic functions.
• Calculus: Derivatives and integrals of quadratic functions are used extensively in calculus to analyze rates of change and areas under curves.

Conclusion: A Versatile Mathematical Expression

The seemingly simple expression "30 increased by 3 times the square of a number" offers a rich mathematical landscape to explore. From solving basic equations to visualizing graphs and understanding advanced concepts, this expression provides a foundation for learning and applying fundamental mathematical principles. Its applications extend far beyond the classroom, touching upon various fields of study and real-world problems. By mastering the concepts related to this expression, you equip yourself with valuable tools for tackling more complex mathematical challenges. The journey from understanding the simple equation to grasping its deeper implications is a testament to the power and beauty of mathematics. Remember to always approach mathematical problems methodically, breaking them down into smaller, manageable steps. This ensures accuracy and a deeper understanding of the underlying principles.