(x^2-5x+4)(x^2-9)=

2 min read Jun 17, 2024

Factoring and Solving (x^2-5x+4)(x^2-9)

This expression represents the product of two quadratic expressions. To fully understand it, we can factor each part and then multiply the results.

Factoring the Quadratic Expressions

(x^2 - 5x + 4): This expression factors into (x - 1)(x - 4). We find this by looking for two numbers that add up to -5 (the coefficient of the x term) and multiply to 4 (the constant term).
(x^2 - 9): This expression is a difference of squares, which factors into (x + 3)(x - 3).

Multiplying the Factored Expressions

Now, we have:

(x - 1)(x - 4)(x + 3)(x - 3)

Finding the Solutions (Roots)

To find the solutions, we set the entire expression equal to zero:

(x - 1)(x - 4)(x + 3)(x - 3) = 0

For this product to equal zero, at least one of the factors must be zero. Therefore, the solutions are:

x = 1
x = 4
x = -3
x = 3

Conclusion

By factoring and multiplying the two quadratic expressions, we have found that (x^2-5x+4)(x^2-9) is equivalent to (x - 1)(x - 4)(x + 3)(x - 3). This factored form reveals the four solutions or roots of the expression, which are x = 1, x = 4, x = -3, and x = 3.