Understanding the Discriminant: When b²-4ac is Greater Than 0, It Indicates Two Distinct Real Solutions for a Quadratic Equation

If b²-4ac > 0

If b²-4ac > 0, it means that the quadratic equation of the form ax² + bx + c = 0 has two distinct real solutions

If b²-4ac > 0, it means that the quadratic equation of the form ax² + bx + c = 0 has two distinct real solutions.

To understand this, let’s break it down:

– A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
– The discriminant, denoted as D, is the term inside the square root in the quadratic formula: D = b² – 4ac.
– The discriminant determines the nature of the solutions of the quadratic equation.

When the discriminant, b² – 4ac, is greater than 0, it means that it’s a positive value. This indicates that there are two distinct real solutions for the quadratic equation.

For example, consider the equation x² – 4x + 3 = 0.

Here, a = 1, b = -4, and c = 3.

To find the discriminant: D = b² – 4ac
D = (-4)² – 4(1)(3)
D = 16 – 12
D = 4

Since the discriminant D is greater than 0, we can conclude that there are two distinct real solutions for the quadratic equation x² – 4x + 3 = 0.

To find the solutions, we can use the quadratic formula:

x = (-b ± √(b² – 4ac)) / (2a)

Substituting the values:

x = (-(-4) ± √(16 – 4(1)(3))) / (2(1))
x = (4 ± √(16 – 12)) / 2
x = (4 ± √4) / 2
x = (4 ± 2) / 2

We get two solutions:

x₁ = (4 + 2) / 2 = 6 / 2 = 3
x₂ = (4 – 2) / 2 = 2 / 2 = 1

Therefore, the quadratic equation x² – 4x + 3 = 0 has two distinct real solutions: x₁ = 3 and x₂ = 1.

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