Question

Consider a quadratic expression, say \\[ x^{2}-x-2 \\] Describing where this quadratic is positive (that is, greater than 0 ). involves describing a set of numbers that are either less than the smaller root (which is -1 ) or greater than the larger root (which is +2 ). Write a \(\mathrm{C}++\) Boolean expression that is true when this formula has positive values.

Short answer

Answer: The quadratic expression \(x^2-x-2\) takes positive values when \(x<-1\) or \(x>2\). The C++ boolean expression to represent this is: ```cpp bool positiveQuadraticValue(double x) { return (x < -1) || (x > 2); } ```

Step-by-step solution

Find the roots of the quadratic equation

In this case, the given quadratic equation is \(x^2-x-2\), and it is already factored as \((x+1)(x-2)\). So, its roots are -1 and +2.

Determine the intervals for which the quadratic expression is positive

The given quadratic equation is positive between the roots, which are -1 and +2 in this case. Thus, the quadratic expression is positive when \(x<-1\) or \(x>2\).

Write a C++ boolean expression

Based on the intervals where the quadratic expression is positive, we can now write a C++ boolean expression: ```cpp bool positiveQuadraticValue(double x) { return (x < -1) || (x > 2); } ``` This function will return true if the given value of x results in the quadratic expression being positive, and false otherwise.

Key concepts

Understanding the Quadratic Equation

The quadratic equation is a fundamental concept that often appears in mathematics and related disciplines. It can be represented in the standard form as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients and \( a \) is not zero, ensuring the equation is indeed quadratic.

A crucial aspect of studying quadratic equations is finding their roots or the solutions to the equation. These are the values of \( x \) that make the equation equal to zero. In the context of our problem, the quadratic expression \( x^2 - x - 2 \) is factored as \( (x + 1)(x - 2) \), which gives us the roots -1 and 2.

To understand where the quadratic expression is positive or negative, we analyze the sign of the product of its factors in the various intervals determined by the roots. The expression is positive outside the interval formed by the roots, that is, when \( x < -1 \) or \( x > 2 \). This knowledge is not only useful in pure mathematics but also in various practical problems such as physics, engineering, and economics, where the relationships expressed by quadratic equations can model complex phenomena.

Programming Problem Solving Skills

When it comes to programming problem solving, it's essential to first understand the mathematical or logical problem you're trying to solve before jumping into code. Breaking down the problem into smaller steps can make it much easier to translate into a programming language.

In the case of our quadratic expression problem, the steps involved identifying the roots and determining where the expression is positive. Problem-solving skills involve systematic thinking, which in programming translates to writing functions or algorithms that resolve each part of the problem. For instance, after analyzing the math, a programmer knows that the problem can be solved by a function that takes a value of \( x \), and then uses comparative logic to return the result. The key here is to create structured, clear, and maintainable code that can be easily understood and modified if needed.

Effective problem solving in programming also entails anticipating potential errors or edge cases and handling them accordingly, which can involve implementing additional conditional checks or input validation.

Conditional Logic in C++

Conditional statements are at the core of many programming tasks, enabling software to make decisions. In C++, such logic is often implemented using the 'if' and 'else' keywords, as well as boolean expressions that resolve to either 'true' or 'false'.

The boolean expression for our quadratic equation problem takes advantage of the logical OR operator '||'. This operator allows the function to return 'true' if at least one of the specified conditions is met. The expression \( (x < -1) || (x > 2) \) confirms whether the given value of \( x \) falls outside the interval formed by the roots of the quadratic equation.

Understanding conditional logic is crucial in programming because it provides a way to execute code based on dynamic conditions, which can change depending on the user input, data received, or any other contextual factor. It empowers the program to react differently under varying circumstances, thus making the program more robust and versatile.