To solve the quadratic equation [tex]\(z^2 - z = 1\)[/tex], we start by bringing all terms to one side to set the equation to zero:
[tex]\[ z^2 - z - 1 = 0. \][/tex]
Next, we solve this quadratic equation using the quadratic formula, which is given by:
[tex]\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \][/tex]
where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are the coefficients from the quadratic equation [tex]\(az^2 + bz + c = 0\)[/tex]. In this case, [tex]\(a = 1\)[/tex], [tex]\(b = -1\)[/tex], and [tex]\(c = -1\)[/tex].
Substitute these values into the quadratic formula:
[tex]\[ z = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-1)}}{2(1)}. \][/tex]
Simplify inside the square root and solve step-by-step:
1. Simplify [tex]\( -(-1) \)[/tex]:
[tex]\[ z = \frac{1 \pm \sqrt{1 + 4}}{2}. \][/tex]
2. Simplify [tex]\(1 + 4\)[/tex]:
[tex]\[ z = \frac{1 \pm \sqrt{5}}{2}. \][/tex]
So, the exact values of [tex]\(z\)[/tex] that make the equation [tex]\(z^2 - z = 1\)[/tex] true are:
[tex]\[ z = \frac{1 - \sqrt{5}}{2} \quad \text{and} \quad z = \frac{1 + \sqrt{5}}{2}. \][/tex]
These solutions match option A. Therefore, the correct answer is:
A. [tex]\( z = \frac{1 \pm \sqrt{5}}{2} \)[/tex].
