Verified Answer :

  • Start with the equation \(x^2 - 2x = 2\).
  • Add the square of half the coefficient of \(x\), which is \((-2/2)^2 = 1\), to both sides to complete the square: \(x^2 - 2x + 1 = 3\).
  • Rewrite the left side as a perfect square: \((x - 1)^2 = 3\).
  • Solve for \(x\) by taking the square root: \(x - 1 = \pm \sqrt{3}\), resulting in \(x = 1 \pm \sqrt{3}\).
  • Final answer: \(\boxed{1 - \sqrt{3}, 1 + \sqrt{3}}\).

Explanation

  1. Problem Analysis

We are given the quadratic equation \(x^2 - 2x = 2\). The objective is to solve this equation for \(x\) using the method of completing the square. Completing the square is a technique used to transform a quadratic expression of the form \(x^2 + bx\) into a perfect square binomial \((x + \frac{b}{2})^2 - (\frac{b}{2})^2\). This allows us to isolate \(x\) by taking the square root of both sides.

  1. Completing the Square

To complete the square for \(x^2 - 2x = 2\), we focus on the coefficient of \(x\), which is \(-2\). The rule for completing the square is to add \(\left(\frac{b}{2}\right)^2\) to both sides of the equation, where \(b\) is the coefficient of the linear term \(x\).
Here, \(b = -2\), so we calculate:
\[\left(\frac{-2}{2}\right)^2 = (-1)^2 = 1.\]
Adding \(1\) to both sides gives:
\[x^2 - 2x + 1 = 2 + 1.\]
This simplifies to:
\[x^2 - 2x + 1 = 3.\]

  1. Solving for x

The left side of our equation, \(x^2 - 2x + 1\), is now a perfect square trinomial, which factors as \((x - 1)^2\). Substituting this into our equation gives:
\[(x - 1)^2 = 3.\]
To solve for \(x\), take the square root of both sides, keeping in mind that we must account for both the positive and negative roots:
\[x - 1 = \pm\sqrt{3}.\]
Adding \(1\) to both sides results in the solutions:
\[x = 1 \pm\sqrt{3}.\]
Thus, the solutions are \(x = 1 + \sqrt{3}\) and \(x = 1 - \sqrt{3}\).

  1. Final Answer

The exact solutions to the quadratic equation \(x^2 - 2x = 2\) are \(1 - \sqrt{3}\) and \(1 + \sqrt{3}\). The solution set is \(\{1 - \sqrt{3}, 1 + \sqrt{3}\}\).
\[\boxed{1 - \sqrt{3}, 1 + \sqrt{3}}\]

Examples

Completing the square is a fundamental tool in algebra, often used to derive the quadratic formula or to analyze functions. For instance, if you are modeling the trajectory of a ball, the quadratic equation describing its height can be converted into vertex form (like \((x-h)^2 + k\)) to easily find the maximum height and the time it takes to reach that height. This method simplifies complex expressions and makes it easier to visualize and solve for specific variables.

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