You can determine the number of solutions a quadratic equation has by examining the discriminant, which is the part of the quadratic formula under the square root: b2 - 4ac.
Here's a breakdown of how the discriminant relates to the number of solutions:
| Discriminant (b2 - 4ac) | Number of Real Solutions | Explanation |
|---|---|---|
| Positive (b2 - 4ac > 0) | Two Solutions | The square root of a positive number yields two real values (one positive and one negative), resulting in two distinct solutions for x. |
| Zero (b2 - 4ac = 0) | One Solution | The square root of zero is zero. The quadratic formula simplifies to x = -b/2a, resulting in only one solution. This also means the vertex of the parabola lies on the x-axis. |
| Negative (b2 - 4ac < 0) | No Real Solutions | The square root of a negative number is not a real number. Therefore, there are no real solutions for x. The solutions are complex numbers. The parabola does not intersect the x-axis. |
Example:
Consider the quadratic equation ax2 + bx + c = 0. The discriminant helps us understand the nature of the roots without actually solving the equation.
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Equation: x2 + 4x + 4 = 0
- a = 1, b = 4, c = 4
- Discriminant: b2 - 4ac = (4)2 - 4(1)(4) = 16 - 16 = 0
- Since the discriminant is zero, there is exactly one real solution.
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Equation: x2 + 2x + 10 = 0
- a = 1, b = 2, c = 10
- Discriminant: b2 - 4ac = (2)2 - 4(1)(10) = 4 - 40 = -36
- Since the discriminant is negative, there are no real solutions.
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Equation: x2 + 5x + 6 = 0
- a = 1, b = 5, c = 6
- Discriminant: b2 - 4ac = (5)2 - 4(1)(6) = 25 - 24 = 1
- Since the discriminant is positive, there are two real solutions.
