To determine the nature of the roots of the quadratic equation [tex]\( x^2 + x + 1 = 0 \)[/tex], we will analyze the discriminant of this equation.
The general form of a quadratic equation is:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
For the given equation [tex]\( x^2 + x + 1 = 0 \)[/tex], we identify the coefficients as:
[tex]\[ a = 1 \][/tex]
[tex]\[ b = 1 \][/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 values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into this formula, we get:
[tex]\[ \Delta = 1^2 - 4 \cdot 1 \cdot 1 \][/tex]
[tex]\[ \Delta = 1 - 4 \][/tex]
[tex]\[ \Delta = -3 \][/tex]
The value of the discriminant helps us determine the nature of the roots of the quadratic equation:
- If [tex]\( \Delta > 0 \)[/tex], the roots are real and distinct.
- If [tex]\( \Delta = 0 \)[/tex], the roots are real and equal.
- If [tex]\( \Delta < 0 \)[/tex], the roots are not-real (complex).
In this problem, we have:
[tex]\[ \Delta = -3 \][/tex]
Since the discriminant is less than zero ([tex]\( \Delta < 0 \)[/tex]), it indicates that the roots of the quadratic equation [tex]\( x^2 + x + 1 = 0 \)[/tex] are not-real (complex).
Thus, the correct answer is:
(D) not-real
