State all possiblities for the number of solutions a quadratic equation can have in one cycle.
ax^2 + bx + c = y
To find the solutions you let y = 0.
x^2 + 6x + 9 = 0
x^2 + 7x + 12 = 0
x^2 - 4 = 0
x^2 + 4 = 0
Solve these and you will have your answer.
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To determine the number of solutions a quadratic equation can have in one cycle, we need to consider the discriminant of the equation. The discriminant can be found by using the formula:
Discriminant (D) = b^2 - 4ac
Here, ‘a’, ‘b’, and ‘c’ represent the coefficients of the quadratic equation in the form: ax^2 + bx + c = 0.
Now, based on the value of the discriminant, we can have three possibilities for the number of solutions:
1. D > 0: If the discriminant is greater than zero, then the quadratic equation has two distinct real solutions. In other words, it intersects the x-axis at two different points. This occurs when the graph of the equation crosses the x-axis at two points, indicating two solutions.
2. D = 0: If the discriminant is equal to zero, then the quadratic equation has one real solution. In this case, the equation intersects the x-axis at only one point. This happens when the graph touches the x-axis at a single point, suggesting only one solution.
3. D < 0: If the discriminant is less than zero, then the quadratic equation has no real solutions. In this scenario, the graph of the equation does not intersect the x-axis at any point. Instead, it remains either entirely above or below the x-axis. However, it is worth noting that there can still be complex solutions involving imaginary numbers, but there are no real solutions.
By considering these possibilities, we can determine the potential number of solutions a quadratic equation may have in one cycle.