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To solve the problem of determining which quadratic has a different discriminant from the others, we will calculate the discriminants for each quadratic equation and compare them.

The discriminant \(\Delta\) of a quadratic equation in the form \( ax^2 + bx + c = 0 \) is given by the formula:
\[
\Delta = b^2 - 4ac
\]
Now, let's compute the discriminant for each of the given quadratics:

### Option a: \( 3x^2 + x - 2 \)
Here, \(a = 3\), \(b = 1\), and \(c = -2\).
\[
\Delta_a = 1^2 - 4(3)(-2) = 1 + 24 = 25
\]

### Option b: \( 4x^2 - 12x + 9 \)
Here, \(a = 4\), \(b = -12\), and \(c = 9\).
\[
\Delta_b = (-12)^2 - 4(4)(9) = 144 - 144 = 0
\]

### Option c: \( x^2 - 4x + 4 \)
Here, \(a = 1\), \(b = -4\), and \(c = 4\).
\[
\Delta_c = (-4)^2 - 4(1)(4) = 16 - 16 = 0
\]

### Conclusion:
- \(\Delta_a = 25\)
- \(\Delta_b = 0\)
- \(\Delta_c = 0\)

The discriminants of options b and c are the same, while the discriminant of option a is different. Therefore, the answer is **option a**.

Would you like further details or clarification on this solution?

Here are 5 questions to expand your understanding:
1. What does the discriminant tell us about the roots of a quadratic equation?
2. Can a quadratic equation have a negative discriminant? What does it mean?
3. How would you factor a quadratic with a discriminant of zero?
4. How do the values of \(a\), \(b\), and \(c\) affect the shape of a quadratic graph?
5. Can a quadratic equation have only one real solution?

**Tip:** The discriminant can help you quickly determine the nature of the roots (real or complex) without fully solving the equation.

Which of the following quadratics has a different discriminant from the other two? Options: a) 3x^2 + x - 2, b) 4x^2 - 12x + 9, c) x^2 - 4x + 4.

Learn how to compare the discriminants of three quadratic equations: 3x^2 + x - 2, 4x^2 - 12x + 9, and x^2 - 4x + 4. This problem demonstrates how to compute discriminants to determine the nature of quadratic roots. Ideal for students in Grades 9-11.

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