Find the value of x if `(14x-4):(8x-1)=(3x+8):(9x+5) `

Options

A. 1
B. `1/2`
C. `3/4`
D. None of these

Answer

To solve the equation \((14x-4):(8x-1)=(3x+8):(9x+5)\), we can set up the equation using cross multiplication. Here’s how to do it step by step:

### Step 1: Set up the equation
From the given ratio, we can write:
\[ \frac{14x - 4}{8x - 1} = \frac{3x + 8}{9x + 5} \]

### Step 2: Cross multiply
Cross multiplying gives us:
\[ (14x - 4)(9x + 5) = (3x + 8)(8x - 1) \]

### Step 3: Expand both sides
Now we will expand both sides of the equation.

**Left Side:**
\[ (14x - 4)(9x + 5) = 14x \cdot 9x + 14x \cdot 5 - 4 \cdot 9x - 4 \cdot 5 \]
\[ = 126x^2 + 70x - 36x - 20 \]
\[ = 126x^2 + 34x - 20 \]

**Right Side:**
\[ (3x + 8)(8x - 1) = 3x \cdot 8x + 3x \cdot (-1) + 8 \cdot 8x + 8 \cdot (-1) \]
\[ = 24x^2 - 3x + 64x - 8 \]
\[ = 24x^2 + 61x - 8 \]

### Step 4: Set the equation to zero
Now we set both sides equal to each other:
\[ 126x^2 + 34x - 20 = 24x^2 + 61x - 8 \]

Subtract \(24x^2 + 61x - 8\) from both sides:
\[ 126x^2 - 24x^2 + 34x - 61x - 20 + 8 = 0 \]
\[ 102x^2 - 27x - 12 = 0 \]

### Step 5: Solve the quadratic equation
Now we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 102\), \(b = -27\), and \(c = -12\).

First, calculate the discriminant:
\[ b^2 - 4ac = (-27)^2 - 4(102)(-12) \]
\[ = 729 + 4896 = 5625 \]

Now, apply the quadratic formula:
\[ x = \frac{-(-27) \pm \sqrt{5625}}{2 \cdot 102} \]
\[ = \frac{27 \pm 75}{204} \]

### Step 6: Calculate the two possible values of x
1. \(x = \frac{27 + 75}{204} = \frac{102}{204} = \frac{1}{2}\)
2. \(x = \frac{27 - 75}{204} = \frac{-48}{204} = -\frac{8}{34} = -\frac{4}{17}\)

### Final Answer
Thus, the values of \(x\) are:
\[ x = \frac{1}{2} \quad \text{or} \quad x = -\frac{4}{17} \]