1.1.2. $$ \underbrace{x^{2}-x=5} $$ (correct

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To solve the equation $$ x^{2} - x = 5 $$, we will follow these steps:

1. **Rearrange the equation**: Move all terms to one side to set the equation to zero.
2. **Solve the resulting quadratic equation**.

Let's start with step 1:

$$
x^{2} - x - 5 = 0
$$

Now, we will solve the quadratic equation $$ x^{2} - x - 5 = 0 $$ using the quadratic formula, which is given by:

$$
x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
$$

In our equation, $$ a = 1 $$, $$ b = -1 $$, and $$ c = -5 $$.

Now, let's calculate the discriminant $$ b^{2} - 4ac $$ and then find the values of $$ x $$.

I'll perform these calculations now.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(-1\right)^{2}-4\times 1\times \left(-5\right)$$
- step1: Evaluate the power:
  $$1-4\times 1\times \left(-5\right)$$
- step2: Multiply the terms:
  $$1+20$$
- step3: Add the numbers:
  $$21$$
Solve the equation $$ x^2 - x - 5 = 0 $$.
Solve the quadratic equation by following steps:
- step0: Solve using the quadratic formula:
  $$x^{2}-x-5=0$$
- step1: Solve using the quadratic formula:
  $$x=\frac{1\pm \sqrt{\left(-1\right)^{2}-4\left(-5\right)}}{2}$$
- step2: Simplify the expression:
  $$x=\frac{1\pm \sqrt{21}}{2}$$
- step3: Separate into possible cases:
  $$\begin{align}&x=\frac{1+\sqrt{21}}{2}\&x=\frac{1-\sqrt{21}}{2}\end{align}$$
- step4: Rewrite:
  $$x_{1}=\frac{1-\sqrt{21}}{2},x_{2}=\frac{1+\sqrt{21}}{2}$$
The discriminant $$ b^{2} - 4ac $$ is calculated as follows:

$$
(-1)^{2} - 4 \cdot 1 \cdot (-5) = 1 + 20 = 21
$$

Since the discriminant is positive, we will have two real solutions for the quadratic equation.

Now, applying the quadratic formula:

$$
x = \frac{-(-1) \pm \sqrt{21}}{2 \cdot 1} = \frac{1 \pm \sqrt{21}}{2}
$$

Thus, the solutions are:

$$
x_{1} = \frac{1 - \sqrt{21}}{2}, \quad x_{2} = \frac{1 + \sqrt{21}}{2}
$$

These are the two solutions to the equation $$ x^{2} - x = 5 $$.
The solutions to the equation $$ x^{2} - x = 5 $$ are $$ x = \frac{1 - \sqrt{21}}{2} $$ and $$ x = \frac{1 + \sqrt{21}}{2} $$.

1.1.2. \( \underbrace{x^{2}-x=5} \) (correct

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