If one zero of the quadratic polynomial `x^2 - 5x - 6` is 6 then find the other zero

Answer

To find the other zero of the quadratic polynomial \(x^2 - 5x - 6\) given that one zero is 6, we can use the properties of the zeros of a quadratic polynomial.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given polynomial is \(x^2 - 5x - 6\). Here, we can identify:
- \(a = 1\) (coefficient of \(x^2\))
- \(b = -5\) (coefficient of \(x\))
- \(c = -6\) (constant term)

2. **Use the sum of the zeros**:
The sum of the zeros of a quadratic polynomial \(ax^2 + bx + c\) is given by the formula:
\[ \text{Sum of zeros} = -\frac{b}{a} \]
Substituting the values of \(b\) and \(a\):
\[ \text{Sum of zeros} = -\frac{-5}{1} = 5 \]

3. **Set up the equation**:
Let the other zero be \(\alpha\). Since one zero is given as 6, we can write:
\[ 6 + \alpha = 5 \]

4. **Solve for \(\alpha\)**:
Rearranging the equation gives:
\[ \alpha = 5 - 6 = -1 \]

5. **Conclusion**:
Therefore, the other zero of the polynomial \(x^2 - 5x - 6\) is \(-1\).

### Verification using the product of the zeros:

1. **Use the product of the zeros**:
The product of the zeros of a quadratic polynomial is given by:
\[ \text{Product of zeros} = \frac{c}{a} \]
Substituting the values of \(c\) and \(a\):
\[ \text{Product of zeros} = \frac{-6}{1} = -6 \]

2. **Set up the equation**:
The product of the zeros can also be expressed as:
\[ 6 \times \alpha = -6 \]

3. **Solve for \(\alpha\)**:
Rearranging gives:
\[ \alpha = \frac{-6}{6} = -1 \]

This confirms that the other zero is indeed \(-1\).