Discuss the sign of the following expressions when x is real
i) `x^(2)-5x+6` ii) `x^(2)-x+3`

Answer

To discuss the sign of the given expressions when \( x \) is real, we will analyze each expression step by step.

### i) Expression: \( x^2 - 5x + 6 \)

**Step 1: Factor the quadratic expression.**
We need to factor \( x^2 - 5x + 6 \).

To factor, we look for two numbers that multiply to \( 6 \) (the constant term) and add to \( -5 \) (the coefficient of \( x \)). The numbers \( -2 \) and \( -3 \) fit this requirement.

Thus, we can factor the expression as:
\[ x^2 - 5x + 6 = (x - 2)(x - 3) \]

**Step 2: Find the roots of the equation.**
Setting the factors equal to zero gives us the roots:
\[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \]
\[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \]

**Step 3: Analyze the sign of the expression.**
We will use a number line to analyze the sign of the expression \( (x - 2)(x - 3) \).

- Choose test points in the intervals:
- \( (-\infty, 2) \): Let \( x = 0 \) → \( (0 - 2)(0 - 3) = (−2)(−3) = 6 \) (positive)
- \( (2, 3) \): Let \( x = 2.5 \) → \( (2.5 - 2)(2.5 - 3) = (0.5)(−0.5) = -0.25 \) (negative)
- \( (3, \infty) \): Let \( x = 4 \) → \( (4 - 2)(4 - 3) = (2)(1) = 2 \) (positive)

**Step 4: Conclusion on the sign.**
The sign of \( x^2 - 5x + 6 \) is:
- Positive for \( x < 2 \)
- Negative for \( 2 < x < 3 \)
- Positive for \( x > 3 \)

### ii) Expression: \( x^2 - x + 3 \)

**Step 1: Check the discriminant.**
We will use the quadratic formula to determine the nature of the roots. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac = (-1)^2 - 4(1)(3) = 1 - 12 = -11 \]

Since the discriminant is negative, this means that the quadratic has no real roots.

**Step 2: Analyze the sign of the expression.**
Since the leading coefficient (the coefficient of \( x^2 \)) is positive (1), the parabola opens upwards. Therefore, since there are no real roots, the expression \( x^2 - x + 3 \) is always positive for all real values of \( x \).

**Step 3: Conclusion on the sign.**
The sign of \( x^2 - x + 3 \) is:
- Positive for all \( x \in \mathbb{R} \).

### Summary of Signs:
- For \( x^2 - 5x + 6 \):
- Positive for \( x < 2 \)
- Negative for \( 2 < x < 3 \)
- Positive for \( x > 3 \)

- For \( x^2 - x + 3 \):
- Positive for all \( x \in \mathbb{R} \)