A line segment AB of length 2 m is divided at a point C into two parts such that `AC^(2) = AB xx CB` . Find the length of CB .

Answer

To solve the problem, we need to find the length of segment CB given the conditions of the problem. Let's break it down step by step.

### Step 1: Define the segments
Let:
- AC = x (length of segment AC)
- CB = 2 - x (since the total length AB is 2 m)

### Step 2: Set up the equation
According to the problem, we have the equation:
\[ AC^2 = AB \times CB \]
Substituting the values we defined:
\[ x^2 = 2 \times (2 - x) \]

### Step 3: Simplify the equation
Expanding the right side:
\[ x^2 = 2(2 - x) \]
\[ x^2 = 4 - 2x \]

### Step 4: Rearrange the equation
Rearranging the equation gives us:
\[ x^2 + 2x - 4 = 0 \]

### Step 5: Apply the quadratic formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation \( x^2 + 2x - 4 = 0 \):
- a = 1
- b = 2
- c = -4

Substituting these values into the formula:
\[ x = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1} \]
\[ x = \frac{-2 \pm \sqrt{4 + 16}}{2} \]
\[ x = \frac{-2 \pm \sqrt{20}}{2} \]
\[ x = \frac{-2 \pm 2\sqrt{5}}{2} \]
\[ x = -1 \pm \sqrt{5} \]

### Step 6: Determine the valid solution for x
Since length cannot be negative, we take:
\[ x = -1 + \sqrt{5} \]

### Step 7: Find the length of CB
Now, we can find CB:
\[ CB = 2 - x = 2 - (-1 + \sqrt{5}) \]
\[ CB = 2 + 1 - \sqrt{5} \]
\[ CB = 3 - \sqrt{5} \]

### Final Answer
The length of CB is \( 3 - \sqrt{5} \) meters.

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