Find the discriminant of the quadratic equation `(p + 3)x^(2) - (5 - p)x + 1 = 0 ` and hence determine the value of p for which the roots are real and distinct.

Answer

To find the discriminant of the quadratic equation \((p + 3)x^{2} - (5 - p)x + 1 = 0\) and determine the values of \(p\) for which the roots are real and distinct, we will follow these steps:

### Step 1: Identify coefficients
The standard form of a quadratic equation is \(Ax^2 + Bx + C = 0\). From the given equation:
- \(A = p + 3\)
- \(B = -(5 - p) = p - 5\)
- \(C = 1\)

### Step 2: Write the formula for the discriminant
The discriminant \(D\) of a quadratic equation is given by:
\[ D = B^2 - 4AC \]

### Step 3: Substitute the coefficients into the discriminant formula
Substituting the values of \(A\), \(B\), and \(C\) into the discriminant formula:
\[ D = (p - 5)^2 - 4(p + 3)(1) \]

### Step 4: Expand and simplify the expression
Now, we will expand and simplify the expression for \(D\):
\[ D = (p - 5)^2 - 4(p + 3) \]
\[ = (p^2 - 10p + 25) - (4p + 12) \]
\[ = p^2 - 10p + 25 - 4p - 12 \]
\[ = p^2 - 14p + 13 \]

### Step 5: Set the discriminant greater than zero for real and distinct roots
For the roots to be real and distinct, the discriminant must be greater than zero:
\[ p^2 - 14p + 13 > 0 \]

### Step 6: Factor the quadratic inequality
Next, we will factor the quadratic:
\[ p^2 - 14p + 13 = (p - 1)(p - 13) \]
Thus, we need to solve:
\[ (p - 1)(p - 13) > 0 \]

### Step 7: Determine the intervals for the inequality
To find the intervals where this product is positive, we identify the roots, which are \(p = 1\) and \(p = 13\). We will test the intervals:
1. \(p < 1\)
2. \(1 < p < 13\)
3. \(p > 13\)

- For \(p < 1\): Choose \(p = 0\) → \((0 - 1)(0 - 13) = 1 \cdot 13 > 0\) (True)
- For \(1 < p < 13\): Choose \(p = 5\) → \((5 - 1)(5 - 13) = 4 \cdot (-8) < 0\) (False)
- For \(p > 13\): Choose \(p = 14\) → \((14 - 1)(14 - 13) = 13 \cdot 1 > 0\) (True)

### Step 8: Conclusion
The solution to the inequality is:
\[ p < 1 \quad \text{or} \quad p > 13 \]
Thus, the values of \(p\) for which the roots of the quadratic equation are real and distinct are \(p < 1\) or \(p > 13\).

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