the value of p for the equation (p+6)x^2 + (p+3)x + 1 =0 has equal roots is

To find the value of \( p \) for which the quadratic equation \( (p+6)x^2 + (p+3)x + 1 = 0 \) has equal roots, we use the condition for equal roots of a quadratic equation, which is when the discriminant (\( \Delta \)) is zero.

For a general quadratic equation \( ax^2 + bx + c = 0 \), the discriminant is given by:

\[
\Delta = b^2 - 4ac
\]

Here, we have:
- \( a = p+6 \)
- \( b = p+3 \)
- \( c = 1 \)

Now, apply these values to the discriminant formula:

\[
\Delta = (p+3)^2 - 4(p+6)(1)
\]

Setting \( \Delta = 0 \) for equal roots:

\[
(p+3)^2 - 4(p+6) = 0
\]

Expanding both terms:

\[
(p+3)^2 = p^2 + 6p + 9
\]
\[
4(p+6) = 4p + 24
\]

Substituting these into the equation:

\[
p^2 + 6p + 9 - 4p - 24 = 0
\]

Simplify:

\[
p^2 + 2p - 15 = 0
\]

Now, solve this quadratic equation:

\[
p^2 + 2p - 15 = 0
\]

Factoring the equation:

\[
(p + 5)(p - 3) = 0
\]

Thus, the solutions are:

\[
p = -5 \quad \text{or} \quad p = 3
\]

So, the values of \( p \) for which the quadratic equation has equal roots are \( p = -5 \) or \( p = 3 \).

Would you like further explanation or additional details on this? Here are some related questions to explore:

1. What are the properties of the discriminant for quadratic equations?
2. How do we determine if a quadratic equation has distinct, real, or complex roots?
3. Can you explain the process of factoring a quadratic equation?
4. What is the geometric interpretation of equal roots for a parabola?
5. How does the discriminant affect the nature of roots?

**Tip:** Always check the discriminant first when solving quadratic equations to know the nature of the roots.

the value of p for the equation (p+6)x^2 + (p+3)x + 1 = 0 has equal roots is

Learn how to find the values of p for which the quadratic equation (p+6)x^2 + (p+3)x + 1 = 0 has equal roots. This detailed explanation covers the discriminant formula, factoring, and algebraic solutions, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation