(x-3)^2-(x-3)(3-x^2)=0

2 min read Jun 17, 2024

Solving the Equation (x-3)^2 - (x-3)(3-x^2) = 0

This article will guide you through solving the equation (x-3)^2 - (x-3)(3-x^2) = 0. We'll break down the process step-by-step, making it easy to understand.

Step 1: Factoring out a common factor

Notice that both terms in the equation share a common factor of (x-3). We can factor this out:

(x-3)[(x-3) - (3-x^2)] = 0

Step 2: Simplifying the expression within the brackets

Let's simplify the expression inside the brackets:

(x-3)[x-3 - 3 + x^2] = 0 (x-3)[x^2 + x - 6] = 0

Step 3: Factoring the quadratic expression

The expression within the brackets is a quadratic equation. We can factor it further:

(x-3)(x+3)(x-2) = 0

Step 4: Setting each factor equal to zero

For the product of factors to be zero, at least one of the factors must equal zero. Therefore, we set each factor equal to zero and solve for x:

x - 3 = 0 => x = 3
x + 3 = 0 => x = -3
x - 2 = 0 => x = 2

Conclusion

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