Question

4x² - 4x - 1 = 0
the solution set is { }.
(type an exact answer, using radicals as needed. use integers or fractions for any numbers in the expression. use a comma to separate answers

Explanation:

Step1: Identify quadratic coefficients

For $ax^2+bx+c=0$, here $a=4$, $b=-4$, $c=-1$

Step2: Apply quadratic formula

Quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
Substitute values:
$x=\frac{-(-4)\pm\sqrt{(-4)^2-4(4)(-1)}}{2(4)}$

Step3: Simplify discriminant

Calculate $b^2-4ac$:
$\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}$

Step4: Simplify the expression

$x=\frac{4\pm4\sqrt{2}}{8}=\frac{1\pm\sqrt{2}}{2}$

Answer:

$\frac{1+\sqrt{2}}{2}, \frac{1-\sqrt{2}}{2}$