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The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by the formula [tex]\( \Delta = b^2 - 4ac \)[/tex]. The value of the discriminant determines the nature of the roots of the quadratic equation:
1. If [tex]\( \Delta > 0 \)[/tex], the equation has two distinct real roots.
2. If [tex]\( \Delta = 0 \)[/tex], the equation has exactly one real root (a repeated root).
3. If [tex]\( \Delta < 0 \)[/tex], the equation has two complex roots (no real roots).
Given the options for the discriminant:
- [tex]\( -9 \)[/tex]
- [tex]\( -7 \)[/tex]
- [tex]\( 7 \)[/tex]
- [tex]\( 9 \)[/tex]
The values [tex]\( -9 \)[/tex] and [tex]\( -7 \)[/tex] indicate that the quadratic equation will have two complex roots.
The values [tex]\( 7 \)[/tex] and [tex]\( 9 \)[/tex] indicate that the quadratic equation will have two distinct real roots.