Understanding the Quadratic Equation | The Role of the Discriminant and Real Solutions

If b²-4ac > 0

If we have the expression b²-4ac > 0, this is actually a condition that relates to the solutions of a quadratic equation

If we have the expression b²-4ac > 0, this is actually a condition that relates to the solutions of a quadratic equation. Let’s consider a quadratic equation in the standard form: ax² + bx + c = 0.

In this equation, a, b, and c are coefficients, and x is the variable we are solving for. The quadratic formula can be used to find the solutions of this equation:

x = (-b ± √(b²-4ac))/(2a)

The expression b²-4ac is called the discriminant, denoted by the symbol Δ (delta). It is used to determine the nature of the solutions of the quadratic equation.

If the discriminant is greater than zero (b²-4ac > 0), it means that the quadratic equation has two distinct real solutions. In other words, there are two different values of x that satisfy the equation and make it true. These solutions lie on the real number line and can be expressed as real numbers.

For example, if we have the quadratic equation x² – 4x + 3 = 0, we can determine the discriminant:

Δ = b²-4ac = (-4)² – 4(1)(3) = 16 – 12 = 4

Since the discriminant (4) is greater than zero, we know that this quadratic equation has two distinct real solutions.

Using the quadratic formula, we can find these solutions:

x = (4 ± √4)/(2*1)
x = (4 ± 2)/2

This gives us the solutions x = 3 and x = 1. So, the quadratic equation x² – 4x + 3 = 0 has two distinct real solutions, which are 3 and 1.

More Answers:
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Understanding the Discriminant of Quadratic Equations | Nature and Solutions
Understanding the Nature of Quadratic Equations | Exploring Complex Solutions and Negative Discriminants

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