Question

Comment on the sign of the quadratic expression \(x^{2}-5 x+6\) for all \(x \in R\). (1) \(x^{2}-5 x+6 \geq 0\) when \(2 \leq x \leq 3\) and (2) \(x^{2}-5 x+6 \leq 0\) when \(2 \leq x \leq 3\) and \(x^{2}-5 x+6<0\) when \(x<2\) or \(x>3 \quad x^{2}-5 x+6>0\) when \(x<2\) or \(x>3\) (3) \(\mathrm{x}^{2}-5 \mathrm{x}+6 \leq 0\) when \(-1 \leq \mathrm{x} \leq 6\) and (4) \(x^{2}-5 x+6 \geq 0\) when \(-1 \leq x \leq 6\) and \(\mathrm{x}^{2}-5 \mathrm{x}+6>0\) when \(\mathrm{x}<-1\) or \(\mathrm{x}>6 \quad \mathrm{x}^{2}-5 \mathrm{x}+6<0\) when \(\mathrm{x}<-1\) or \(\mathrm{x}>6\)

Short answer

Answer: The quadratic expression \(x^{2}-5x+6\) is positive when \(x<2\) or \(x>3\) and non-positive (i.e., less than or equal to zero) when \(2 \leq x \leq 3\).

Step-by-step solution

Find the roots of the quadratic equation

First, we need to solve the equation \(x^{2}-5x+6=0\). To do this, we can try to factor the quadratic expression or use the quadratic formula. It looks like factoring is possible in this case: \((x-2)(x-3)=0\). Therefore, the roots of the equation are \(x=2\) and \(x=3\).

Analyze intervals between roots

Since we have found the roots of the quadratic equation, we can now analyze the intervals (inequalities) between these roots to determine the sign of the expression. We have three intervals to consider: \(x < 2\), \(2 \leq x \leq 3\), and \(x > 3\). By observing the quadratic expression \((x-2)(x-3)\), we can determine the signs for these intervals: 1. If \(x < 2\), then \((x-2) < 0\) and \((x-3) < 0\). Therefore, \((x-2)(x-3) > 0\). 2. If \(2 \leq x \leq 3\), then \((x-2) \geq 0\) and \((x-3) \leq 0\). Therefore, \((x-2)(x-3) \leq 0\). 3. If \(x > 3\), then \((x-2) > 0\) and \((x-3) > 0\). Therefore, \((x-2)(x-3) > 0\).

Summarize the sign of the quadratic expression

In summary, the sign of the quadratic expression \(x^{2}-5x+6\) for all \(x \in R\) can be described as follows: 1. \(x^{2}-5x+6 > 0\) when \(x<2\) or \(x>3\). 2. \(x^{2}-5x+6 \leq 0\) when \(2 \leq x \leq 3\).

Key concepts

Roots of Quadratic Equations

To fully understand a quadratic expression, finding its roots is essential. Roots are the values of the variable that make the expression equal to zero. For our quadratic expression, \(x^2 - 5x + 6\), finding the roots involves solving the equation \(x^2 - 5x + 6 = 0\). This can be done using factoring, a common method suitable for simple quadratics. In this case, the expression factors neatly into \((x-2)(x-3) = 0\). This reveals the roots as \(x = 2\) and \(x = 3\).

Remember, the roots split the number line into intervals which are crucial for analyzing the behavior of the quadratic expression. When a quadratic is factored, the expression changes sign around these roots, influencing the intervals to consider in further analysis. Understanding these roots helps clarify whether the quadratic expression is positive, negative, or zero at different intervals.

Sign of Quadratic Expressions

The sign of a quadratic expression, like \(x^2 - 5x + 6\), depends on the roots and the intervals they create on the number line. Here, we examine the intervals between and beyond the roots \(x = 2\) and \(x = 3\).

1. **Interval \(x < 2\):** Both factors \((x-2)\) and \((x-3)\) are negative. A negative times a negative results in a positive, so the expression \((x-2)(x-3) > 0\).
2. **Interval \(2 \leq x \leq 3\):** Here, \((x-2)\) is zero or positive and \((x-3)\) is zero or negative. Since the zero presence constrains the expression, the product \((x-2)(x-3) \leq 0\).
3. **Interval \(x > 3\):** Both factors \((x-2)\) and \((x-3)\) are positive. The result is a positive product: \((x-2)(x-3) > 0\).

Knowing the signs of the expression during these intervals helps in graphing and problem-solving, especially in solving inequalities with quadratic expressions.

Inequalities in Quadratic Functions

Understanding inequalities in quadratic functions is key in determining where the function is greater than or less than zero. Given our expression \(x^2 - 5x + 6\), we can use the signs obtained from the previous analysis to set inequalities.

- **\(x^2 - 5x + 6 > 0\):** This inequality holds when \(x < 2\) or \(x > 3\). In these intervals, the quadratic expression is positive, which might represent solutions where the function is above the \(x\)-axis in a graph.
- **\(x^2 - 5x + 6 \leq 0\):** This is true when \(2 \leq x \leq 3\). Here the expression is either zero or negative. It indicates that the parabola lies on or below the \(x\)-axis between these roots.

Quadratic inequalities provide information about where on the number line a quadratic expression holds certain properties like positivity or negativity. This is crucial in real-world applications where constraints must align with practical requirements, relying heavily on understanding the intervals gathered from the roots and their sign behavior.