Solving Quadratic Equations with the Quadratic Formula

Study Notes

Solving Quadratic Equations with the Quadratic Formula

Quadratic equations are second-degree polynomial equations, featuring variables raised to the power of 2. They're represented in the general form of (ax^2 + bx + c = 0), where (a), (b), and (c) are constants. Unlike linear equations, which have only one solution, quadratic equations typically have two solutions, called roots (or zeros).

Solving quadratic equations can be achieved through various methods, but the quadratic formula is a reliable and straightforward approach. Here's how it works:

The Quadratic Formula

The quadratic formula is a convenient way to find the roots of a quadratic equation. It's derived from the fact that the roots of a quadratic equation are the values of (x) that make the equation equal to zero. The formula is given by:

[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}]

Here, the variables (a), (b), and (c) are the coefficients of the quadratic equation.

Using the Quadratic Formula

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

Write the quadratic equation in the general form (ax^2 + bx + c = 0).
Substitute the values of (a), (b), and (c) into the quadratic formula.
Simplify and solve for (x) by finding the two roots: (x_1) and (x_2).

Examples

Consider the quadratic equation (x^2 + 3x + 2 = 0). To find the roots using the quadratic formula:

Write the equation in the general form: (x^2 + 3x + 2 = 0).
Determine the values of (a = 1), (b = 3), and (c = 2).
Substitute these values into the quadratic formula:

[x = \frac{-3 \pm \sqrt{3^2 - 4(1)(2)}}{2(1)}]

[x = \frac{-3 \pm \sqrt{9 - 8}}{2}]

Simplify and solve for (x):

[x = \frac{-3 \pm \sqrt{1}}{2} = \frac{-3 \pm 1}{2}]
The two roots are (x_1 = \frac{-3 + 1}{2} = \frac{-2}{2} = -1) and (x_2 = \frac{-3 - 1}{2} = \frac{-4}{2} = -2).

Discriminant

The discriminant, denoted as (D), is the value under the square root in the quadratic formula. It determines the nature of the roots:

If (D > 0), the roots are real and distinct.
If (D = 0), the roots are real and equal (a double root or a repeated root).
If (D < 0), the roots are complex (imaginary).

In the previous example, the discriminant is (D = (3)^2 - 4(1)(2) = 9 - 8 = 1), and the roots are real and distinct.

Solving quadratic equations using the quadratic formula is a reliable and efficient method. However, it's essential to remember that not all quadratic equations can be solved using this method, for example, equations with complex roots or equations that result in irrational expressions. In these cases, other methods may be more suitable, such as completing the square, the quadratic trinomial factoring method, or numerical techniques like the Babylonian method or the Newton-Raphson method.

Description

Explore how to solve quadratic equations using the quadratic formula, a reliable method for finding the roots of second-degree polynomial equations. Learn about the quadratic formula, its application, and how to determine the nature of roots using the discriminant. Practice solving quadratic equations step by step and understand the concept of real, distinct, equal, and complex roots.