Understanding Quadratic Equations: Two Distinct Real Solutions and the Discriminant

If b²-4ac > 0

In mathematics, when the quadratic equation is written in the form of ax^2 + bx + c = 0, a, b, and c are coefficients, and x is the variable

In mathematics, when the quadratic equation is written in the form of ax^2 + bx + c = 0, a, b, and c are coefficients, and x is the variable. The discriminant is given by b² – 4ac.

If the discriminant b² – 4ac is greater than zero (b² – 4ac > 0), then it means that the quadratic equation has two distinct real solutions. In other words, the equation intersects the x-axis at two different points.

To understand this concept, let’s break it down further:

1. When b² – 4ac > 0, it implies that the value inside the square root (√) in the quadratic formula is positive.
√(b² – 4ac) > 0

2. Recall the quadratic formula:
x = (-b ± √(b² – 4ac)) / (2a)

3. Since √(b² – 4ac) > 0, both the numerator and denominator will be greater than zero, such that:
(-b ± √(b² – 4ac)) > 0
(2a) > 0

4. Taking the positive case where (-b + √(b² – 4ac)) > 0 yields:
-b + √(b² – 4ac) > 0

5. Solving for x in the quadratic formula, considering the positive case, will give one solution:
x = (-b + √(b² – 4ac)) / (2a)

6. Similarly, considering the negative case where (-b – √(b² – 4ac)) > 0 yields:
-b – √(b² – 4ac) > 0

7. Solving for x in the quadratic formula, considering the negative case, will give another solution:
x = (-b – √(b² – 4ac)) / (2a)

So, when b² – 4ac is greater than zero, the quadratic equation has two distinct real solutions, which can be found using the quadratic formula.

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