Solve x²-2x+1=9: Using the Abbreviated Multiplication Formula

We will solve the quadratic equation $x^2 - 2x + 1 = 9$ using the square of a binomial formula.

Firstly, let's recognize that the left side of the equation forms a perfect square:

$x^2 - 2x + 1 \equiv (x - 1)^2$

Therefore, the equation can be rewritten as:

$(x - 1)^2 = 9$

To solve for $x$, take the square root of both sides. Remember to consider both the positive and negative solutions from the square root:

Thus, $x - 1 = \pm 3$

This gives us two separate equations to solve:

• $x - 1 = 3$
• $x - 1 = -3$

Solving each equation for $x$ gives:

• For $x - 1 = 3$:

Add 1 to both sides: $x = 4$

• For $x - 1 = -3$:

Add 1 to both sides: $x = -2$

Therefore, the solutions to the equation are $x = 4$ and $x = -2$.

Comparing these solutions to the given answer choices, we identify the correct choice as:

$x=-2$ or $x=4$

In conclusion, the solutions to the equation are $x = 4$ and $x = -2$.