Question

solve $x^2 - 8x + 5 = 0$ using the completing-the-square method.

$\bigcirc$ $x = -4 \pm \sqrt{11}$

$\bigcirc$ $x = -4 \pm \sqrt{5}$

$\bigcirc$ $x = 4 \pm \sqrt{11}$

$\bigcirc$ $x = 4 \pm \sqrt{5}$

Explanation:

Step1: Isolate the x-terms

Start with the equation \(x^2 - 8x + 5 = 0\). Subtract 5 from both sides to get \(x^2 - 8x = -5\).

Step2: Complete the square

Take half of the coefficient of \(x\) (which is -8), so \(\frac{-8}{2} = -4\), then square it: \((-4)^2 = 16\). Add 16 to both sides of the equation: \(x^2 - 8x + 16 = -5 + 16\).

Step3: Factor the perfect square

The left side factors as \((x - 4)^2\), and the right side simplifies to 11: \((x - 4)^2 = 11\).

Step4: Solve for x

Take the square root of both sides: \(x - 4 = \pm\sqrt{11}\). Then add 4 to both sides: \(x = 4 \pm\sqrt{11}\).

Answer:

\(x = 4 \pm\sqrt{11}\) (corresponding to the option "x = 4 ± √11")