To determine whether a quadratic equation has real solutions, we need to examine its discriminant. For any quadratic equation in standard form ax squared plus bx plus c equals zero, where a is not equal to zero, the discriminant delta is calculated as b squared minus 4ac.
The discriminant value determines the nature of solutions for the quadratic equation. When delta is greater than zero, the equation has two distinct real solutions. When delta equals zero, there is exactly one repeated real solution. When delta is less than zero, the equation has no real solutions, only complex solutions.
We can visualize this concept using parabolas. The red parabola with discriminant greater than zero intersects the x-axis at two points, representing two real solutions. The blue parabola with discriminant equal to zero touches the x-axis at exactly one point. The green parabola with discriminant less than zero does not intersect the x-axis at all, indicating no real solutions.
Let's work through a specific example. Consider the equation 2x squared minus 5x plus 3 equals zero. Here, a equals 2, b equals negative 5, and c equals 3. The discriminant is calculated as negative 5 squared minus 4 times 2 times 3, which equals 25 minus 24, giving us 1. Since delta equals 1, which is greater than zero, this equation has two distinct real solutions.
In summary, the discriminant delta equals b squared minus 4ac is the key tool for determining whether a quadratic equation has real solutions. When delta is greater than or equal to zero, real solutions exist. When delta is less than zero, there are no real solutions. The process is simple: first identify the coefficients a, b, and c, then calculate the discriminant, and finally check if delta is greater than or equal to zero.