Use the quadratic formula to complete the table. To verify your solutions, graph the equations.
| Equation | | Value of Discriminant | Solution Value(s) |
| :---------------- | :-: | :-------------------- | :---------------- |
| [tex]$3x^2 + 4x + 4 = 0$[/tex] | | | |
| 42 + [tex]$2+1+0$[/tex] | | | |
Answer (2)
The first equation has a discriminant of -32, indicating no real solutions. The second equation simplifies to a contradiction with no solutions. Thus, there are no real solutions for either equation.
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For the equation 3 x 2 + 4 x + 4 = 0 , the discriminant is − 32 , and there are no real solutions. For the equation 42 + 2 + 1 + 0 = 0 , there are no solutions, and the discriminant does not exist.
Examples
Quadratic equations are used in various real-life applications, such as calculating the trajectory of a projectile, designing parabolic mirrors, or determining the optimal dimensions for a rectangular enclosure with a given area. Understanding the discriminant helps determine the nature of the solutions, indicating whether a projectile will hit a target, whether a mirror will focus light, or whether a rectangular enclosure is even possible with the given constraints. For example, if an engineer is designing a bridge with a parabolic arch, they need to ensure that the arch's equation has real roots to ensure the bridge is structurally sound.
