Question

completing the square
solve by completing the square.
$x^2 - 4x = 1$
$x = \square \pm \sqrt{\square}$

Explanation:

Step1: Identify the coefficient of x

For the quadratic equation \(x^2 - 4x = 1\), the coefficient of \(x\) is \(-4\). To complete the square, we take half of this coefficient, square it, and add it to both sides. Half of \(-4\) is \(-2\), and squaring \(-2\) gives \((-2)^2 = 4\).

Step2: Add the square to both sides

Add \(4\) to both sides of the equation: \(x^2 - 4x + 4 = 1 + 4\). The left side now factors as a perfect square: \((x - 2)^2 = 5\).

Step3: Solve for x

Take the square root of both sides: \(x - 2 = \pm\sqrt{5}\). Then, add \(2\) to both sides to solve for \(x\): \(x = 2 \pm\sqrt{5}\).

Answer:

\(x = \boldsymbol{2} \pm \sqrt{\boldsymbol{5}}\)