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Solving the Equation (x-3)^2 - 2 = x + 1
This article explores the solution process for the equation (x-3)^2 - 2 = x + 1.
Step 1: Expanding the Equation
First, we need to expand the left side of the equation by using the formula (a-b)^2 = a^2 - 2ab + b^2.
(x-3)^2 - 2 = x + 1 x^2 - 6x + 9 - 2 = x + 1
Step 2: Simplifying the Equation
Next, combine like terms on both sides to simplify the equation.
x^2 - 6x + 7 = x + 1 x^2 - 7x + 6 = 0
Step 3: Factoring the Equation
Now, we factor the quadratic equation. The goal is to find two numbers that multiply to 6 and add to -7. The numbers -6 and -1 satisfy these conditions.
(x - 6)(x - 1) = 0
Step 4: Solving for x
Setting each factor equal to zero, we can solve for x.
x - 6 = 0 or x - 1 = 0 x = 6 or x = 1
Solution
Therefore, the solutions to the equation (x-3)^2 - 2 = x + 1 are x = 6 and x = 1.