Understanding the Quadratic Formula

The quadratic formula is a method used to find the solutions (or roots) of quadratic equations. A quadratic equation is an equation of the form:

Ax² + Bx + C = 0

Where:

• A is the coefficient of x² (the quadratic term),
• B is the coefficient of x (the linear term),
• C is the constant term.

What is the Quadratic Formula?

The quadratic formula is expressed as:

x = (-B ± √(B² – 4AC)) / 2A

Here, the symbols have the following meanings:

• B² – 4AC is called the discriminant. It tells us the nature of the roots.
• √(B² – 4AC) is the square root of the discriminant.
• ± indicates that there are two possible solutions: one where you add the square root and one where you subtract it.

How to Use the Quadratic Formula?

To solve a quadratic equation using the quadratic formula, follow these steps:

1. Identify the values of A, B, and C from your equation.
2. Substitute the values of A, B, and C into the quadratic formula.
3. Calculate the discriminant (B² – 4AC). If the discriminant is positive, the equation has two real roots. If it’s zero, the equation has one real root (the roots are equal). If the discriminant is negative, the equation has no real roots, but two complex (imaginary) roots.
4. Finally, calculate the values of x using both the positive and negative square roots (the ± symbol).

Example

Let’s say you have the equation:

2x² + 3x – 2 = 0

Here, A = 2, B = 3, and C = -2. Plugging these values into the quadratic formula:

x = (-3 ± √(3² – 4(2)(-2))) / 2(2)

x = (-3 ± √(9 + 16)) / 4

x = (-3 ± √25) / 4

x = (-3 ± 5) / 4

The two solutions are:

• x₁ = (-3 + 5) / 4 = 2 / 4 = 0.5
• x₂ = (-3 – 5) / 4 = -8 / 4 = -2

So, the solutions for x are 0.5 and -2.

Why is the Quadratic Formula Important?

The quadratic formula is essential because it provides a straightforward method for solving quadratic equations, regardless of whether the coefficients are integers, fractions, or decimals. It works for all types of quadratic equations, making it a powerful tool in algebra and higher-level mathematics.