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Solve the equation using the quadratic formula. Express your answer in exact, simplified form.



To solve the quadratic equation \( p^2 - 14p + 49 = 0 \) using the quadratic formula, we follow these steps:

1. Identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation \(ax^2 + bx + c = 0\).
   - \(a = 1\)
   - \(b = -14\)
   - \(c = 49\)

2. The quadratic formula is given by:
   \[
   p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]

3. Substitute the values of \(a\), \(b\), and \(c\) into the formula:
   \[
   p = \frac{-(-14) \pm \sqrt{(-14)^2 - 4 \cdot 1 \cdot 49}}{2 \cdot 1}
   \]

4. Simplify under the square root:
   \[
   p = \frac{14 \pm \sqrt{196 - 196}}{2}
   \]
   \[
   p = \frac{14 \pm \sqrt{0}}{2}
   \]
   \[
   p = \frac{14 \pm 0}{2}
   \]

5. Since \(\sqrt{0} = 0\), the solution simplifies to:
   \[
   p = \frac{14}{2} = 7
   \]

Therefore, the solution set is \(\{7\}\).

Would you like more details or have any questions?

Here are some related questions you might consider:
1. How do you derive the quadratic formula?
2. What are the different methods to solve a quadratic equation?
3. Can you show an example of solving a quadratic equation by completing the square?
4. How does the discriminant determine the nature of the roots in a quadratic equation?
5. What is the geometric interpretation of a quadratic equation's solutions?

**Tip:** Always check if the quadratic equation can be factored easily before using the quadratic formula, as factoring can sometimes be quicker.

Learn how to solve the quadratic equation p^2 - 14p + 49 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic formula. Suitable for students in Grades 9-12.

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