Determine whether the value given against each of the qaudratic equation are the roots of the equation.
(ii) `2x^2-7x+6=0 , x=3/2,-2`.

Answer

To determine whether the values \( x = \frac{3}{2} \) and \( x = -2 \) are roots of the quadratic equation \( 2x^2 - 7x + 6 = 0 \), we will substitute each value into the equation and check if the left-hand side equals zero.

### Step 1: Check if \( x = \frac{3}{2} \) is a root

1. Substitute \( x = \frac{3}{2} \) into the equation:
\[ 2\left(\frac{3}{2}\right)^2 - 7\left(\frac{3}{2}\right) + 6 \]

2. Calculate \( \left(\frac{3}{2}\right)^2 \):
\[ \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

3. Substitute this back into the equation:
\[ 2 \cdot \frac{9}{4} - 7 \cdot \frac{3}{2} + 6 \]

4. Simplify \( 2 \cdot \frac{9}{4} \):
\[ = \frac{18}{4} = \frac{9}{2} \]

5. Calculate \( -7 \cdot \frac{3}{2} \):
\[ = -\frac{21}{2} \]

6. Now substitute these values back into the equation:
\[ \frac{9}{2} - \frac{21}{2} + 6 \]

7. Combine the fractions:
\[ = \frac{9 - 21}{2} + 6 = -\frac{12}{2} + 6 = -6 + 6 = 0 \]

Since the left-hand side equals zero, \( x = \frac{3}{2} \) is a root of the equation.

### Step 2: Check if \( x = -2 \) is a root

1. Substitute \( x = -2 \) into the equation:
\[ 2(-2)^2 - 7(-2) + 6 \]

2. Calculate \( (-2)^2 \):
\[ (-2)^2 = 4 \]

3. Substitute this back into the equation:
\[ 2 \cdot 4 - 7 \cdot (-2) + 6 \]

4. Simplify \( 2 \cdot 4 \):
\[ = 8 \]

5. Calculate \( -7 \cdot (-2) \):
\[ = 14 \]

6. Now substitute these values back into the equation:
\[ 8 + 14 + 6 \]

7. Combine the values:
\[ = 8 + 14 + 6 = 28 \]

Since the left-hand side does not equal zero, \( x = -2 \) is not a root of the equation.

### Conclusion
- \( x = \frac{3}{2} \) is a root of the quadratic equation \( 2x^2 - 7x + 6 = 0 \).
- \( x = -2 \) is not a root of the quadratic equation.