(x-a)(x-b) Formula Answer

4 min read Jun 17, 2024

Understanding the (x-a)(x-b) Formula

The formula (x-a)(x-b) is a fundamental concept in algebra, specifically when dealing with quadratic equations. This formula represents the factored form of a quadratic expression, and it's essential for solving equations and understanding the properties of parabolas.

What does the Formula Represent?

The formula represents the product of two binomials, (x-a) and (x-b). These binomials represent linear expressions where 'a' and 'b' are constants. When you multiply these two binomials, you get a quadratic expression:

(x-a)(x-b) = x² - (a+b)x + ab

How to use the Formula

You can use this formula in two main ways:

1. Expanding a Factored Expression:

If you have an expression in the form (x-a)(x-b), you can use the formula to expand it into a quadratic expression. Simply follow the pattern:

x²: The product of the 'x' terms from each binomial.
-(a+b)x: The sum of the products of the 'x' term from one binomial and the constant term from the other.
ab: The product of the constant terms from each binomial.

2. Factoring a Quadratic Expression:

If you have a quadratic expression in the form ax² + bx + c, you can use the formula to factor it into the form (x-a)(x-b). To do this, you need to find the values of 'a' and 'b' that satisfy the following conditions:

a + b = -b
ab = c

You can use various techniques like the quadratic formula or factorization methods to find these values.

Example

Let's say we have the expression (x-2)(x+3). We can use the formula to expand this:

x²: x * x = x²
-(2-3)x: -2 * x + x * 3 = -2x + 3x = x
2 * -3: 2 * -3 = -6

Therefore, (x-2)(x+3) = x² + x - 6

Importance

The (x-a)(x-b) formula is crucial because:

Solves Quadratic Equations: By factoring quadratic equations using this formula, we can find the roots (solutions) of the equations.
Graphing Parabolas: The formula helps determine the x-intercepts of a parabola, which are the points where the parabola intersects the x-axis.
Simplifying Expressions: It can be used to simplify more complex algebraic expressions.

In conclusion, the (x-a)(x-b) formula is a powerful tool in algebra that provides a straightforward way to work with quadratic expressions and gain a deeper understanding of their properties.