Solving the Quadratic Equation: x2 3x - 18 0

When dealing with quadratic equations, a common approach is to factor the equation if possible. The equation in question, (x^2 3x - 18 0), is a perfect example where this method can be effectively applied.

Understanding the Equation

The given equation (x^2 3x - 18 0) is a quadratic equation. It does not have an equals sign in the form of a linear expression but is set to zero. To solve for (x), various methods are available, with the factoring method being one of the most straightforward when the equation can be easily decomposed into its factors.

Factoring the Quadratic Expression

Let's look at the factored form of this equation. The first step is to find two numbers that multiply to (-18) (the constant term) and add up to (3) (the coefficient of (x)). The factors of (-18) that satisfy these conditions are (6) and (-3), since (6 times -3 -18) and (6 (-3) 3).

Step-by-Step Solution

Divide the middle term (3x) into two parts, (6x) and (-3x):
x2 6x - 3x - 18 0 Group the terms to form factor pairs:
(x2 6x) (-3x - 18) 0 Factor out the common factors from each group:
x(x 6) - 3(x 6) 0 Factor out the common binomial factor ((x 6)):
(x 6)(x - 3) 0

Solving for x

The resulting factors provide the solutions to the quadratic equation:

For ((x 6) 0):
x -6 For ((x - 3) 0):
x 3

Therefore, the solutions to the equation (x^2 3x - 18 0) are x -6 and x 3.

Verifying the Solutions

To ensure that these solutions are correct, you can substitute them back into the original equation:

Substitute x -6

(x^2 3x - 18 (-6)^2 3(-6) - 18 36 - 18 - 18 0)

Substitute x 3

(x^2 3x - 18 (3)^2 3(3) - 18 9 9 - 18 0)

Both substitutions confirm that the solutions are correct.

Additional Methods for Solving Quadratic Equations

Completing the Square: An alternative method involves transforming the quadratic equation into a perfect square trinomial. This method is particularly useful when the coefficient of (x^2) is 1 and the middle term is even. Quadratic Formula: The general formula for solving any quadratic equation (ax^2 bx c 0) is given by:
x (-b pm sqrt{b^2 - 4ac}) / 2a

Both of these methods can be useful in different scenarios and are valuable for mastering the solution of quadratic equations.

By understanding and applying the factoring method and other techniques for solving quadratic equations, you can handle a wide variety of algebraic problems with confidence.