Write all the values of p for which the quadratic equation `x^(2) + px + 16 = 0` has equal roots . Find the roots of the equation so obtained .

Answer

To solve the quadratic equation \(x^2 + px + 16 = 0\) for the values of \(p\) that yield equal roots, we will follow these steps:

### Step 1: Understand the condition for equal roots
For a quadratic equation \(ax^2 + bx + c = 0\) to have equal roots, the discriminant must be zero. The discriminant \(D\) is given by:
\[ D = b^2 - 4ac \]
In our equation, \(a = 1\), \(b = p\), and \(c = 16\).

### Step 2: Set up the discriminant equation
Substituting the values into the discriminant formula:
\[ D = p^2 - 4(1)(16) = p^2 - 64 \]
For the roots to be equal, we set the discriminant to zero:
\[ p^2 - 64 = 0 \]

### Step 3: Solve for \(p\)
Now, we solve the equation:
\[ p^2 = 64 \]
Taking the square root of both sides gives:
\[ p = \pm 8 \]
Thus, the values of \(p\) for which the quadratic equation has equal roots are \(p = 8\) and \(p = -8\).

### Step 4: Find the roots of the equations
Now we will find the roots for both cases of \(p\).

#### Case 1: \(p = 8\)
Substituting \(p = 8\) into the original equation:
\[ x^2 + 8x + 16 = 0 \]
This can be factored as:
\[ (x + 4)^2 = 0 \]
Thus, the root is:
\[ x = -4 \]

#### Case 2: \(p = -8\)
Substituting \(p = -8\) into the original equation:
\[ x^2 - 8x + 16 = 0 \]
This can also be factored as:
\[ (x - 4)^2 = 0 \]
Thus, the root is:
\[ x = 4 \]

### Final Answers
The values of \(p\) for which the quadratic equation has equal roots are \(p = 8\) and \(p = -8\). The roots of the equations are:
- For \(p = 8\), the root is \(x = -4\).
- For \(p = -8\), the root is \(x = 4\).

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