To determine the nature of the roots for the quadratic equation [tex]\(5x^2 + 3x = -8\)[/tex], we first rewrite it in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex].
1. Rewrite the equation:
[tex]\[5x^2 + 3x + 8 = 0\][/tex]
2. Identify the coefficients:
[tex]\[a = 5\][/tex]
[tex]\[b = 3\][/tex]
[tex]\[c = 8\][/tex]
3. Calculate the discriminant [tex]\(\Delta\)[/tex]. The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] 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]:
[tex]\[\Delta = 3^2 - 4 \cdot 5 \cdot 8\][/tex]
[tex]\[\Delta = 9 - 160\][/tex]
[tex]\[\Delta = -151\][/tex]
4. Determine the nature of the roots based on the discriminant:
- If [tex]\(\Delta > 0\)[/tex], the roots are real and different.
- If [tex]\(\Delta = 0\)[/tex], the roots are real and equal.
- If [tex]\(\Delta < 0\)[/tex], the roots are imaginary.
Since [tex]\(\Delta = -151\)[/tex] and [tex]\(\Delta < 0\)[/tex], the roots of the quadratic equation [tex]\(5x^2 + 3x + 8 = 0\)[/tex] are imaginary.
Thus, the nature of the roots of [tex]\(5 x^2 + 3 x = -8\)[/tex] is:
Imaginary roots.
