To factor the quadratic equation x² + 3x + 4 = 0, we need to look for two numbers that multiply to the constant term (4) and add up to the coefficient of the x term (3). However, we can quickly see that there are no two real numbers that meet these criteria, since the product of any two positive numbers that multiply to 4 will not add up to 3.
This indicates that the quadratic does not factor nicely into real numbers. To verify, we can use the quadratic formula:
x = (-b ± √(b² – 4ac)) / (2a)
Here, a = 1, b = 3, and c = 4. Plugging these values into the formula:
x = (-3 ± √(3² – 4(1)(4))) / (2(1))
Calculating the discriminant:
3² – 4(1)(4) = 9 – 16 = -7
Since the discriminant is negative, this confirms that the equation has no real roots, and thus cannot be factored into real linear factors. Instead, we can express the roots in terms of complex numbers, but that’s a different topic. Therefore, the given equation does not have a correct factored form in the realm of real numbers.