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 problem, we need to find the value of \( x \) such that the determinants are equal. We start with the given determinants:

\[ |(2,5),(8,x^2)| = |(6,5),(8,3)| \]

### Step 1: Calculate the determinant on the left side

The determinant of a \( 2 \times 2 \) matrix \( |(a,b),(c,d)| \) is given by \( ad - bc \).

For the left determinant:

\[ |(2,5),(8,x^2)| = 2 \cdot x^2 - 5 \cdot 8 \]

Calculating this gives:

\[ 2x^2 - 40 \]

### Step 2: Calculate the determinant on the right side

Now, we calculate the determinant on the right side:

\[ |(6,5),(8,3)| = 6 \cdot 3 - 5 \cdot 8 \]

Calculating this gives:

\[ 18 - 40 = -22 \]

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

Now, we set the two determinants equal to each other:

\[ 2x^2 - 40 = -22 \]

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

To solve for \( x^2 \), we first add 40 to both sides:

\[ 2x^2 = -22 + 40 \]

This simplifies to:

\[ 2x^2 = 18 \]

Now, divide both sides by 2:

\[ x^2 = 9 \]

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

Taking the square root of both sides gives:

\[ x = \pm 3 \]

### Conclusion

Thus, the values of \( x \) are:

\[ x = 3 \quad \text{or} \quad x = -3 \]