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The problem is asking to find the value of \( m \) such that the given quadratic equation has equal roots. The equation provided is:

\[
x^2 + (m - 3)x + m = 0
\]

### Condition for Equal Roots

For a quadratic equation \( ax^2 + bx + c = 0 \) to have equal roots, the discriminant must be zero. The discriminant \( \Delta \) is given by:

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

For the given equation \( x^2 + (m - 3)x + m = 0 \), we have:
- \( a = 1 \)
- \( b = (m - 3) \)
- \( c = m \)

Now, applying the discriminant formula:

\[
\Delta = (m - 3)^2 - 4(1)(m)
\]

Set the discriminant equal to zero, since the equation must have equal roots:

\[
(m - 3)^2 - 4m = 0
\]

Now, solve this equation step by step.

### Step 1: Expand the terms
\[
(m - 3)^2 = m^2 - 6m + 9
\]

Substitute this into the equation:
\[
m^2 - 6m + 9 - 4m = 0
\]

### Step 2: Simplify the equation
\[
m^2 - 10m + 9 = 0
\]

### Step 3: Solve the quadratic equation
Use the quadratic formula \( m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = -10 \), and \( c = 9 \):

\[
m = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(9)}}{2(1)}
\]
\[
m = \frac{10 \pm \sqrt{100 - 36}}{2}
\]
\[
m = \frac{10 \pm \sqrt{64}}{2}
\]
\[
m = \frac{10 \pm 8}{2}
\]

Thus, the two solutions for \( m \) are:
\[
m = \frac{10 + 8}{2} = 9 \quad \text{or} \quad m = \frac{10 - 8}{2} = 1
\]

### Final Answer:
The values of \( m \) for which the quadratic equation has equal roots are \( m = 9 \) or \( m = 1 \).

Would you like further details or have any other questions?

Here are 5 related questions you can explore:
1. How does the discriminant affect the nature of the roots of a quadratic equation?
2. What are the steps to solve quadratic equations using the quadratic formula?
3. Can you explain the geometric interpretation of equal roots in a quadratic equation?
4. How can we factorize a quadratic equation if we know it has equal roots?
5. What is the difference between solving quadratic equations by factorization and using the quadratic formula?

**Tip:** Always verify if a quadratic equation is factorable before resorting to the quadratic formula—it can save time.

Find the value of m if the given quadratic equation x^2 + (m - 3)x + m = 0 has equal roots.

Learn how to find the value of m for the quadratic equation x^2 + (m - 3)x + m = 0, when the equation has equal roots. This step-by-step solution covers key algebraic concepts such as the discriminant and quadratic formula. Suitable for students in Grades 9-10.

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