When using the quadratic formula, why is only necessary to examine the discriminant to determine if the equation only has real number solutions?

To understand why it is only necessary to examine the discriminant when using the quadratic formula, let's review what the quadratic formula is and how it relates to real number solutions.

The quadratic formula is a tool used to find the solutions of a quadratic equation of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x represents the variable.

The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / 2a

The discriminant is the expression inside the square root of the quadratic formula, which helps determine the nature of the solutions. It is calculated as b^2 - 4ac.

Now, let's consider the three different cases based on the value of the discriminant:

1. If the discriminant is positive (b^2 - 4ac > 0), then the quadratic equation has two distinct real number solutions. This means that the equation crosses the x-axis at two different points.

2. If the discriminant is zero (b^2 - 4ac = 0), then the quadratic equation has one real number solution. This means that the equation touches the x-axis at only one point.

3. If the discriminant is negative (b^2 - 4ac < 0), then the quadratic equation has no real number solutions. This means that the equation does not intersect or touch the x-axis at any real number points.

By examining the value of the discriminant, we can determine the nature of the solutions without actually calculating the solutions themselves. If the discriminant is positive or zero, we know that the quadratic equation has real number solutions. However, if the discriminant is negative, we can immediately conclude that the quadratic equation has no real number solutions.

Therefore, when using the quadratic formula, it is only necessary to examine the discriminant to determine if the equation only has real number solutions, as the value of the discriminant directly indicates the nature of the solutions.