Solution of quadratic equation using Factorization

Based on the sources and our conversation history, solving a quadratic equation using the factoring method is a common technique for finding its roots (solutions).

A quadratic equation is obtained when a quadratic function, of the form ax² + bx + c where a is not equal to 0, is set equal to a value, often 0 for the standard form. The standard form is ax² + bx + c = 0, where a, b, and c are typically considered integers.

The solutions to a quadratic equation are called its roots. For the standard form ax² + bx + c = 0, these roots are the x-values where the associated quadratic function f(x) = ax² + bx + c has a value of 0. These points correspond to the x-intercepts of the graph of the quadratic function. Finding the roots algebraically is equivalent to finding the zeros or x-intercepts of the function.

The factoring method relies on expressing the quadratic polynomial as a product of binomials. If a quadratic function can be written in the form a times (x - p) times (x - q), this is called the intercept form, and p and q are the x-intercepts or roots.

Here are the general steps to solve a quadratic equation by factoring, as described in the sources:

• Write the equation in standard form (equal to 0). This means rearranging the terms so that all terms are on one side and the other side is 0.
• Factor the polynomial. This involves finding two expressions (binomials) whose product is the original quadratic polynomial. The sources illustrate this using examples, sometimes showing how to find factors of the constant term and leading coefficient that add or subtract to the middle term’s coefficient. In simpler cases, it might involve taking out a greatest common factor. Converting between the standard form and the intercept form can be done using methods like the FOIL method (First, Outer, Inner, Last) for multiplying binomials.
• Use the Zero Product Property to set each factor equal to zero. This property states that if the product of two factors is zero, then at least one of the factors must be zero.
• Solve each resulting linear equation. Each binomial factor set equal to zero results in a simple linear equation (e.g., x - p = 0) that can be easily solved for x. The solutions obtained are the roots of the quadratic equation.

The sources provide examples demonstrating this method. For instance, solving x² + 2x = 24 involves rewriting it as x² + 2x - 24 = 0, factoring to get (x + 6)(x - 4) = 0, setting each factor to zero (x + 6 = 0 or x - 4 = 0), and solving to find the roots x = -6 and x = 4.

While factoring is a straightforward method, the sources note it may not always be easy to apply, especially if the factors are not readily visible or if the constant term is not a “nice number”. However, it is particularly helpful when the constant term is zero or the factors are easy to find. This method can also be used as a step in solving other types of equations, such as some exponential equations.