find the discriminant of the quadratic equation 2(p²+q²)x² + 2( p+q)x +1 =0 . Also determine the nature of the roots

We are given the quadratic equation:

\[
2(p^2 + q^2)x^2 + 2(p + q)x + 1 = 0
\]

### Step 1: Identifying the coefficients

The general form of a quadratic equation is:

\[
ax^2 + bx + c = 0
\]

From the given equation, we can identify the coefficients as:

- \( a = 2(p^2 + q^2) \)
- \( b = 2(p + q) \)
- \( c = 1 \)

### Step 2: Formula for the Discriminant

The discriminant \( \Delta \) of a quadratic equation is given by the formula:

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

### Step 3: Substituting the values of \(a\), \(b\), and \(c\)

Using the identified coefficients, we can now substitute them into the discriminant formula:

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

### Step 4: Simplifying the expression

1. First, simplify \( b^2 \):

\[
[2(p + q)]^2 = 4(p + q)^2 = 4(p^2 + 2pq + q^2)
\]

2. Now simplify \( 4ac \):

\[
4[2(p^2 + q^2)](1We are given the quadratic equation:

\[
2(p^2 + q^2)x^2 + 2(p + q)x + 1 = 0
\]

### Step 1: Identifying the coefficients

The general form of a quadratic equation is:

\[
ax^2 + bx + c = 0
\]

From the given equation, we can identify the coefficients as:

- \( a = 2(p^2 + q^2) \)
- \( b = 2(p + q) \)
- \( c = 1 \)

### Step 2: Formula for the Discriminant

The discriminant \( \Delta \) of a quadratic equation is given by the formula:

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

### Step 3: Substituting the values of \(a\), \(b\), and \(c\)

Using the identified coefficients, we can now substitute them into the discriminant formula:

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

### Step 4: Simplifying the expression

1. First, simplify \( b^2 \):

\[
[2(p + q)]^2 = 4(p + q)^2 = 4(p^2 + 2pq + q^2)
\]

2. Now simplify \( 4ac \):

\[
4  = 8(p^2 + q^2)
\]

3. Now, substituting these values into the discriminant formula:

\[
\Delta = 4(p^2 + 2pq + q^2) - 8(p^2 + q^2)
\]

### Step 5: Further simplification

We can now expand and simplify:

\[
\Delta = 4p^2 + 8pq + 4q^2 - 8p^2 - 8q^2
\]

\[
\Delta = -4p^2 + 8pq - 4q^2
\]

\[
\Delta = 4(-p^2 + 2pq - q^2)
\]

\[
\Delta = 4(-(p - q)^2)
\]

Thus, the discriminant is:

\[
\Delta = -4(p - q)^2
\]

### Step 6: Determining the nature of the roots

- If \( \Delta > 0 \), the roots are real and distinct.
- If \( \Delta = 0 \), the roots are real and equal.
- If \( \Delta < 0 \), the roots are complex (non-real).

Since \( \Delta = -4(p - q)^2 \), and \( (p - q)^2 \geq 0 \), the discriminant is always non-positive (\( \Delta \leq 0 \)).

- If \( p = q \), then \( \Delta = 0 \), and the roots are real and equal.
- If \( p \neq q \), then \( \Delta < 0 \), and the roots are complex (non-real).

### Conclusion

- The discriminant is \( \Delta = -4(p - q)^2 \).
- The roots are complex if \( p \neq q \) and real and equal if \( p = q \).

Would you like further details or clarifications? Here are some related questions:

1. How does the discriminant affect the nature of the roots in a quadratic equation?
2. Can the discriminant ever be positive in this equation?
3. What happens if both \(p\) and \(q\) are set to specific values like 0 or 1?
4. How can you derive the roots of this equation using the quadratic formula?
5. What other methods can be used to analyze the nature of roots in complex cases?

**Tip:** Always verify if your quadratic equation is in its simplest form before identifying coefficients.

Find the discriminant of the quadratic equation 2(p²+q²)x² + 2(p+q)x +1 = 0. Also determine the nature of the roots.

Learn how to find the discriminant of the quadratic equation 2(p²+q²)x² + 2(p+q)x + 1 = 0. This step-by-step solution covers key algebraic concepts, including the discriminant formula and its role in determining the nature of roots. Suitable for high school students in Grades 10-12.

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