Final answer:
Factoring a quadratic expression involves finding factors that can be multiplied to produce the original expression, with common methods including grouping and the quadratic formula.
Step-by-step explanation:
Factoring a quadratic expression is the process of breaking it down into simpler terms (factors) that can be multiplied together to give the original quadratic. A quadratic expression is generally of the form ax2 + bx + c = 0, where a, b, and c are constants, and x represents an unknown variable.
To factor a quadratic, one method is to find two numbers that multiply to give ac (the product of a and c) and add to give b (the coefficient of x). These two numbers are used to rewrite the middle term (bx), allowing the expression to be factored by grouping. Another option is to use the quadratic formula for cases that are not easily factorable.
For example, if we are given a quadratic expression 4x2 - 4x - 8, we can factor it by finding two numbers that multiply to -32 (4 * -8) and add up to -4. These numbers are -8 and 4, so we can rewrite the middle term as -8x + 4x and factor by grouping to obtain (2x + 2)(2x - 4), simplifying to 2(x + 1)(2x - 4). If the expression was not factorable, we could apply the quadratic formula, which is x = (-b ± √(b2 - 4ac)) / (2a).