If `x = sqrt(5) + 2`, then the value `(2x^2 - 3x - 2)/(3x^2 - 4x - 3)` is equal to
यदि `x = sqrt(5) + 2`, तो`(2x^2 - 3x - 2)/(3x^2 - 4x - 3)` का मान किसके बराबर होगा?

Options

A. `0.185`
B. `0.525 `
C. `0.625`
D. `0.785 `

Explanation

To find the value of \((2x^2 - 3x - 2)/(3x^2 - 4x - 3)\) when \(x = \sqrt{5} + 2\), we will substitute \(x\) into the expression and simplify it step by step.

### Step 1: Substitute \(x\) into the expression
Given \(x = \sqrt{5} + 2\), we substitute this value into the expression:
\[ \frac{2x^2 - 3x - 2}{3x^2 - 4x - 3} \]

### Step 2: Calculate \(x^2\)
First, we need to calculate \(x^2\):
\[ x^2 = (\sqrt{5} + 2)^2 = (\sqrt{5})^2 + 2 \cdot \sqrt{5} \cdot 2 + 2^2 = 5 + 4\sqrt{5} + 4 = 9 + 4\sqrt{5} \]

### Step 3: Calculate \(2x^2\)
Now, we calculate \(2x^2\):
\[ 2x^2 = 2(9 + 4\sqrt{5}) = 18 + 8\sqrt{5} \]

### Step 4: Calculate \(3x\)
Next, we calculate \(3x\):
\[ 3x = 3(\sqrt{5} + 2) = 3\sqrt{5} + 6 \]

### Step 5: Substitute into the numerator
Now substitute \(2x^2\) and \(3x\) into the numerator:
\[ 2x^2 - 3x - 2 = (18 + 8\sqrt{5}) - (3\sqrt{5} + 6) - 2 \]
Simplifying this:
\[ = 18 + 8\sqrt{5} - 3\sqrt{5} - 6 - 2 = 18 - 6 - 2 + (8\sqrt{5} - 3\sqrt{5}) = 10 + 5\sqrt{5} \]

### Step 6: Calculate \(3x^2\)
Next, we calculate \(3x^2\):
\[ 3x^2 = 3(9 + 4\sqrt{5}) = 27 + 12\sqrt{5} \]

### Step 7: Calculate \(4x\)
Now, calculate \(4x\):
\[ 4x = 4(\sqrt{5} + 2) = 4\sqrt{5} + 8 \]

### Step 8: Substitute into the denominator
Now substitute \(3x^2\) and \(4x\) into the denominator:
\[ 3x^2 - 4x - 3 = (27 + 12\sqrt{5}) - (4\sqrt{5} + 8) - 3 \]
Simplifying this:
\[ = 27 - 8 - 3 + (12\sqrt{5} - 4\sqrt{5}) = 16 + 8\sqrt{5} \]

### Step 9: Form the complete expression
Now we have:
\[ \frac{10 + 5\sqrt{5}}{16 + 8\sqrt{5}} \]

### Step 10: Simplify the fraction
To simplify, we can factor out common terms:
\[ = \frac{5(2 + \sqrt{5})}{8(2 + \sqrt{5})} \]
Cancelling \(2 + \sqrt{5}\) from numerator and denominator:
\[ = \frac{5}{8} \]

### Step 11: Convert to decimal
Finally, converting \(\frac{5}{8}\) to decimal form:
\[ \frac{5}{8} = 0.625 \]

Thus, the value of \((2x^2 - 3x - 2)/(3x^2 - 4x - 3)\) is \(\frac{5}{8}\) or \(0.625\).

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