If `|(2,5),(8,x^2)|=|(6,5),(8,3)|` then find the value of x .

Options

A. `+- 2`
B. `+- 3`
C. 0
D. `-1`

Answer

To solve the equation \( |(2,5),(8,x^2)| = |(6,5),(8,3)| \), we will calculate the determinants of both matrices and set them equal to each other.

### Step 1: Calculate the determinant of the left matrix

The determinant of a 2x2 matrix \( |(a,b),(c,d)| \) is given by the formula:
\[ |A| = ad - bc \]
For the left matrix \( |(2,5),(8,x^2)| \):
- \( a = 2 \)
- \( b = 5 \)
- \( c = 8 \)
- \( d = x^2 \)

So, the determinant is:
\[ |L| = 2 \cdot x^2 - 5 \cdot 8 = 2x^2 - 40 \]

### Step 2: Calculate the determinant of the right matrix

For the right matrix \( |(6,5),(8,3)| \):
- \( a = 6 \)
- \( b = 5 \)
- \( c = 8 \)
- \( d = 3 \)

So, the determinant is:
\[ |R| = 6 \cdot 3 - 5 \cdot 8 = 18 - 40 = -22 \]

### Step 3: Set the determinants equal to each other

Now we set the determinants equal:
\[ 2x^2 - 40 = -22 \]

### Step 4: Solve for \( x^2 \)

Add 40 to both sides:
\[ 2x^2 = -22 + 40 \]
\[ 2x^2 = 18 \]

Now, divide both sides by 2:
\[ x^2 = 9 \]

### Step 5: Find the value of \( x \)

Taking the square root of both sides gives:
\[ x = \pm 3 \]

Thus, the values of \( x \) are \( 3 \) and \( -3 \).

### Final Answer:
The value of \( x \) is \( 3 \) or \( -3 \).

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