(1 point) The exact values of [tex]\(z\)[/tex] that make the equation [tex]\(z^2 - z = 1\)[/tex] true are:

A. [tex]\(z = \frac{1 \pm \sqrt{5}}{2}\)[/tex]
B. [tex]\(z = \frac{1}{2} \pm \sqrt{5}\)[/tex]
C. [tex]\(z = -1, 1\)[/tex]
D. [tex]\(z = 0.6, 1.6\)[/tex]



Answer :

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].