Questions: Solve the equation by completing the square. Give the solutions in exact form and in x^2-4x-2=0 exact form x= decimal form x ≈

Solution Steps

To solve the quadratic equation \(x^2 - 4x - 2 = 0\) by completing the square, we first need to rearrange the equation to isolate the constant term. Then, we add and subtract the square of half the coefficient of \(x\) to complete the square. This will allow us to express the quadratic in the form \((x - p)^2 = q\), from which we can solve for \(x\) by taking the square root of both sides. Finally, we provide the solutions in both exact and decimal forms.

Start with the quadratic equation: \[ x^2 - 4x - 2 = 0 \]

To complete the square, we need to express the quadratic in the form \((x - p)^2 = q\). First, rearrange the equation: \[ x^2 - 4x = 2 \]

Add and subtract \(\left(\frac{b}{2}\right)^2 = \left(\frac{-4}{2}\right)^2 = 4\) to the left side: \[ x^2 - 4x + 4 = 2 + 4 \]

This gives: \[ (x - 2)^2 = 6 \]

Take the square root of both sides: \[ x - 2 = \pm \sqrt{6} \]

Solve for \(x\): \[ x = 2 \pm \sqrt{6} \]

Calculate the decimal approximations: \[ x_1 \approx 2 + \sqrt{6} \approx 4.4495 \] \[ x_2 \approx 2 - \sqrt{6} \approx -0.4495 \]

Final Answer