Number of integral values of m for which roots of quadratic equation (m – 1)x2 + 3x + (m – 3) = 0 lie either side of origin, is -

We are given the quadratic equation:

\[
(m – 1)x^2 + 3x + (m – 3) = 0
\]

For the roots of this quadratic equation to lie on either side of the origin, the product of the roots must be negative. 

### Step 1: Recall the general form of a quadratic equation
The general form of a quadratic equation is:

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

Here, we can identify:
- \( a = m - 1 \)
- \( b = 3 \)
- \( c = m - 3 \)

### Step 2: Use the relation for the product of roots
For a quadratic equation \( ax^2 + bx + c = 0 \), the product of the roots is given by:

\[
\text{Product of roots} = \frac{c}{a}
\]

Thus, for our equation, the product of the roots is:

\[
\frac{m - 3}{m - 1}
\]

### Step 3: Set the condition for the product of roots to be negative
For the roots to lie on either side of the origin, the product of the roots must be negative:

\[
\frac{m - 3}{m - 1} < 0
\]

### Step 4: Solve the inequality
The inequality \( \frac{m - 3}{m - 1} < 0 \) holds when one of \( m - 3 \) or \( m - 1 \) is positive, and the other is negative. Let's solve it.

#### Case 1: \( m - 3 > 0 \) and \( m - 1 < 0 \)
- \( m - 3 > 0 \) implies \( m > 3 \)
- \( m - 1 < 0 \) implies \( m < 1 \)

There is no value of \( m \) that satisfies both \( m > 3 \) and \( m < 1 \), so this case has no solution.

#### Case 2: \( m - 3 < 0 \) and \( m - 1 > 0 \)
- \( m - 3 < 0 \) implies \( m < 3 \)
- \( m - 1 > 0 \) implies \( m > 1 \)

Thus, \( 1 < m < 3 \).

### Step 5: Find the integral values of \( m \)
The integral values of \( m \) that satisfy \( 1 < m < 3 \) are:

\[
m = 2
\]

### Conclusion
There is only **one** integral value of \( m \), which is \( m = 2 \), for which the roots lie on either side of the origin.

---

Do you need more details or have any questions?

Here are 5 related questions to consider:
1. What is the sum of the roots for the given quadratic equation?
2. How does the discriminant of a quadratic equation affect the nature of its roots?
3. For what values of \( m \) will the roots of the given equation be equal?
4. How does the quadratic formula help in solving quadratic equations?
5. Can the roots of this quadratic equation both be positive for some values of \( m \)?

**Tip:** When solving quadratic inequalities, always check the sign of both the numerator and denominator to determine where the inequality holds.

Number of integral values of m for which roots of quadratic equation (m – 1)x^2 + 3x + (m – 3) = 0 lie either side of origin, is -

Explore how to find the integral values of m for which the roots of the quadratic equation (m – 1)x^2 + 3x + (m – 3) = 0 lie on either side of the origin. Understand key algebraic concepts, including the general form of a quadratic equation, product of roots formula, and inequalities. Suitable for Grades 9-11.

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