# How do you find the discriminant of #x^2+4x+3=0# and use it to determine if the equation has one, two real or two imaginary roots?

Where

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To find the discriminant of the quadratic equation (x^2 + 4x + 3 = 0), you use the formula for the discriminant, which is given by (b^2 - 4ac), where (a), (b), and (c) are the coefficients of the quadratic equation (ax^2 + bx + c = 0).

In the equation (x^2 + 4x + 3 = 0), (a = 1), (b = 4), and (c = 3). Substituting these values into the discriminant formula, we get:

[\text{Discriminant} = b^2 - 4ac] [= (4)^2 - 4(1)(3)] [= 16 - 12] [= 4]

Now, to determine the nature of the roots based on the discriminant:

- If the discriminant is greater than 0 ((> 0)), the equation has two distinct real roots.
- If the discriminant is equal to 0 ((= 0)), the equation has one real root (a repeated root).
- If the discriminant is less than 0 ((< 0)), the equation has two imaginary (complex) roots.

In this case, the discriminant is 4, which is greater than 0. Therefore, the equation (x^2 + 4x + 3 = 0) has two distinct real roots.

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