Question

Consider the quadratic expression \\[ x^{2}-4 x+3 \\] Describing where this quadratic is negative involves describing a set of numbers that are simultaneously greater than the smaller root (+1) and less than the larger root \((+3) .\) Write a \(\mathrm{C}++\) Boolean expression that is true when the value of this quadratic is negative.

Short answer

Question: Write a C++ Boolean expression that returns true when the quadratic expression $$x^2 - 4xCC + 3$$ is negative, given that the two roots are +1 and +3. Answer: ```cpp bool isQuadraticNegative(double x) { return (x > 1) && (x < 3); } ```

Step-by-step solution

Find the roots of the quadratic

We are given that the roots of the quadratic expression are +1 and +3. That means when we substitute these values for x, the quadratic equation equals 0.

Write the inequality

We need the values of x for which the quadratic expression is negative. Since the quadratic has roots at +1 and +3, it is negative for values of x between these roots, which can be represented as an inequality: $$1 < x < 3$$

Write the C++ Boolean expression

To represent the inequality in C++ code, we can use a Boolean expression: ```cpp bool isQuadraticNegative(double x) { return (x > 1) && (x < 3); } ``` This function checks whether the given value of x is greater than 1 and less than 3. If both conditions are met, the function returns true, indicating that the quadratic expression is negative for the given value of x.

Key concepts

Roots of Quadratic Equations

Quadratic equations often show up in mathematics, involving expressions of the form \(ax^2 + bx + c = 0\). The roots of a quadratic equation are the solutions for \(x\) that satisfy this equation. For the expression \(x^2 - 4x + 3\), the roots are the values of \(x\) where the expression equals zero. These are found by either factoring, using the quadratic formula, or completing the square.

• Factoring: This method involves rewriting the quadratic expression as a product of two binomials. For example, \(x^2 - 4x + 3\) factors to \((x-1)(x-3)\), giving roots \(x=1\) and \(x=3\).
• Quadratic Formula: The formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) provides another way to find the roots of any quadratic equation.

In this particular expression, you can see that the roots are \(x=1\) and \(x=3\). Identifying these points is crucial since they help determine where the expression changes from positive to negative.

Inequalities

Inequalities are statements that indicate one quantity is larger or smaller than another. In the case of quadratic expressions, they help determine the range of \(x\)-values that make an expression positive or negative. For the expression \(x^2 - 4x + 3\), you are asked when this expression is negative. Between the roots \(x=1\) and \(x=3\), the expression becomes negative.

• Open Interval: This is expressed as \(1 < x < 3\), which is an open interval because it excludes the endpoints where the quadratic is zero.
• Graphical Interpretation: On a graph, this interval corresponds to the section below the \(x\)-axis between the points at \(x=1\) and \(x=3\).

Understanding inequalities helps us discuss not only the solutions to equations but also the behavior of expressions over a range of values.

Boolean Expressions in C++

Boolean expressions are used in programming to make decisions based on conditions. In C++, they evaluate to either "true" or "false" and often involve relational operators like \( >, <, ==, \) etc. For instance, within our context, to determine when the quadratic expression \(x^2 - 4x + 3\) is negative, we can create a Boolean expression.
```cppbool isQuadraticNegative(double x) { return (x > 1) && (x < 3);}```

• Relational Operators: The expression \((x > 1) && (x < 3)\) makes use of the greater than \(>\) and less than \(<\) operators.
• Logical Operator: The "&&" operator is a logical AND that ensures both conditions need to be true for the entire statement to be true.

This function evaluates to "true" only when \(x\) is within the interval \(1 < x < 3\), reflecting the range where the quadratic is negative. Such implementations are fundamental in programming, especially in control structures like loops and conditionals.