Calculate the value of x in the following equation :-
(i)`x^(2)-5x+6=0`
(ii)`x^(2)-9x+20=0`
(iii)` x^(2)-15x+56=0`
(iv)`x^(2)-16x+63=0`

Answer

To solve the equations given in the question, we will use the method of factoring. Let's go through each equation step by step.

### (i) Solve \(x^2 - 5x + 6 = 0\)

1. **Identify the coefficients**: Here, \(a = 1\), \(b = -5\), and \(c = 6\).
2. **Find two numbers that multiply to \(c\) (6) and add to \(b\) (-5)**: The numbers are -2 and -3.
3. **Rewrite the equation**:
\[ x^2 - 2x - 3x + 6 = 0 \]
4. **Factor by grouping**:
\[ x(x - 2) - 3(x - 2) = 0 \]
\[ (x - 2)(x - 3) = 0 \]
5. **Set each factor to zero**:
\[ x - 2 = 0 \quad \text{or} \quad x - 3 = 0 \]
6. **Solve for \(x\)**:
\[ x = 2 \quad \text{or} \quad x = 3 \]

### (ii) Solve \(x^2 - 9x + 20 = 0\)

1. **Identify the coefficients**: \(a = 1\), \(b = -9\), and \(c = 20\).
2. **Find two numbers that multiply to \(c\) (20) and add to \(b\) (-9)**: The numbers are -4 and -5.
3. **Rewrite the equation**:
\[ x^2 - 4x - 5x + 20 = 0 \]
4. **Factor by grouping**:
\[ x(x - 4) - 5(x - 4) = 0 \]
\[ (x - 4)(x - 5) = 0 \]
5. **Set each factor to zero**:
\[ x - 4 = 0 \quad \text{or} \quad x - 5 = 0 \]
6. **Solve for \(x\)**:
\[ x = 4 \quad \text{or} \quad x = 5 \]

### (iii) Solve \(x^2 - 15x + 56 = 0\)

1. **Identify the coefficients**: \(a = 1\), \(b = -15\), and \(c = 56\).
2. **Find two numbers that multiply to \(c\) (56) and add to \(b\) (-15)**: The numbers are -7 and -8.
3. **Rewrite the equation**:
\[ x^2 - 7x - 8x + 56 = 0 \]
4. **Factor by grouping**:
\[ x(x - 7) - 8(x - 7) = 0 \]
\[ (x - 7)(x - 8) = 0 \]
5. **Set each factor to zero**:
\[ x - 7 = 0 \quad \text{or} \quad x - 8 = 0 \]
6. **Solve for \(x\)**:
\[ x = 7 \quad \text{or} \quad x = 8 \]

### (iv) Solve \(x^2 - 16x + 63 = 0\)

1. **Identify the coefficients**: \(a = 1\), \(b = -16\), and \(c = 63\).
2. **Find two numbers that multiply to \(c\) (63) and add to \(b\) (-16)**: The numbers are -7 and -9.
3. **Rewrite the equation**:
\[ x^2 - 7x - 9x + 63 = 0 \]
4. **Factor by grouping**:
\[ x(x - 7) - 9(x - 7) = 0 \]
\[ (x - 7)(x - 9) = 0 \]
5. **Set each factor to zero**:
\[ x - 7 = 0 \quad \text{or} \quad x - 9 = 0 \]
6. **Solve for \(x\)**:
\[ x = 7 \quad \text{or} \quad x = 9 \]

### Summary of Solutions:
- For \(x^2 - 5x + 6 = 0\), \(x = 2\) or \(x = 3\)
- For \(x^2 - 9x + 20 = 0\), \(x = 4\) or \(x = 5\)
- For \(x^2 - 15x + 56 = 0\), \(x = 7\) or \(x = 8\)
- For \(x^2 - 16x + 63 = 0\), \(x = 7\) or \(x = 9\)