The quadratic expression \( x^2 - 3x + 4 \) cannot be factored into real numbers because its discriminant is negative.

To check for factorization, we first calculate the discriminant (\( \Delta \)):

\[
\Delta = b^2 - 4ac
\]
where \( a = 1 \), \( b = -3 \), and \( c = 4 \).

\[
\Delta = (-3)^2 - 4(1)(4) = 9 - 16 = -7
\]

Since the discriminant is negative (\( \Delta = -7 \)), the expression does not have real roots and cannot be factored over the real numbers. However, it can be factored using complex numbers as:

\[
x^2 - 3x + 4 = (x - \frac{3}{2} + i\sqrt{\frac{7}{4}})(x - \frac{3}{2} - i\sqrt{\frac{7}{4}})
\]

If you are interested in factoring this using complex numbers, let me know, and I can provide the details.

Would you like to go into more details or have any questions?

Here are 5 related questions to explore further:
1. How do you solve quadratic equations using the quadratic formula?
2. What are the different methods for factoring quadratics?
3. How does the discriminant determine the nature of the roots?
4. Can you graph the quadratic \(x^2 - 3x + 4\) to see why it doesn't cross the x-axis?
5. How do you factor quadratic expressions when complex numbers are involved?

**Tip:** When the discriminant of a quadratic equation is negative, the roots are complex, and you can use the quadratic formula to find them!

Explore the step-by-step process of determining whether the quadratic expression x^2 - 3x + 4 can be factored. Learn why its discriminant is negative, how it leads to complex roots, and how the expression can be factored over the complex numbers. This topic is suitable for Grades 9-12.

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