A Guide to the Quadratic Formula

A quadratic equation is a second-degree polynomial equation in a single variable x with the standard form ax² + bx + c = 0, where 'a', 'b', and 'c' are coefficients and 'a' is not equal to zero. The solutions to this equation are called roots or zeros.

The Quadratic Formula Explained

The quadratic formula is a powerful and reliable method for solving any quadratic equation. It states that the roots (x) of the equation are given by:

x = ( -b ± √(b² - 4ac) ) / 2a

The `x²` term in the equation is a power, a concept you can explore with our Exponent Calculator.

The Discriminant: Understanding the Nature of the Roots

The expression inside the square root, Δ = b² - 4ac, is called the discriminant. The value of the discriminant is critical because it tells us the number and type of roots the equation has without having to solve the full formula:

• If Δ > 0, there are two distinct real roots. The parabola (the graph of a quadratic equation) intersects the x-axis at two different points.
• If Δ = 0, there is exactly one real root (also called a double root). The vertex of the parabola touches the x-axis at one point.
• If Δ < 0, there are two complex roots (conjugate pairs). The parabola does not intersect the x-axis at all.

Step-by-Step Manual Calculation Example

Let's solve the equation 2x² - 5x - 3 = 0:

1. Identify coefficients:
a = 2, b = -5, c = -3

2. Calculate the discriminant (Δ):
Δ = b² - 4ac = (-5)² - 4(2)(-3) = 25 - (-24) = 49. Since Δ > 0, we expect two real roots.

3. Apply the quadratic formula:
x = ( -(-5) ± √49 ) / (2 * 2)
x = ( 5 ± 7 ) / 4

4. Find the two roots:
x₁ = (5 + 7) / 4 = 12 / 4 = 3
x₂ = (5 - 7) / 4 = -2 / 4 = -0.5

For more intricate calculations beyond quadratic equations, you can always use a versatile tool like our Scientific Calculator. Or explore our full suite of Math Calculators for various needs.