To determine the nature of the roots of the quadratic equation [tex]\(3x^2 + 4x - 1 = 0\)[/tex], we need to analyze the discriminant of the equation.
The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are constants. For the given equation:
[tex]\[ a = 3 \][/tex]
[tex]\[ b = 4 \][/tex]
[tex]\[ c = -1 \][/tex]
The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the given values into the discriminant formula:
[tex]\[ \Delta = 4^2 - 4(3)(-1) \][/tex]
[tex]\[ \Delta = 16 + 12 \][/tex]
[tex]\[ \Delta = 28 \][/tex]
The value of the discriminant [tex]\(\Delta\)[/tex] is 28.
The nature of the roots of the quadratic equation is determined by the value of the discriminant:
1. If [tex]\(\Delta > 0\)[/tex], the roots are real and distinct.
2. If [tex]\(\Delta = 0\)[/tex], the roots are real and repeated (coinciding).
3. If [tex]\(\Delta < 0\)[/tex], the roots are imaginary (complex but not real).
Since the discriminant [tex]\(\Delta = 28\)[/tex] is greater than 0, we conclude that the roots of the quadratic equation [tex]\(3x^2 + 4x - 1 = 0\)[/tex] are real and distinct.
Therefore, the nature of the roots is Real and distinct.
