Find the zeroes of the quadratic polynomial `x^(2)-36 ` and verify the relationship between the zeroes and the coefficients .

Answer

To find the zeroes of the quadratic polynomial \(x^2 - 36\) and verify the relationship between the zeroes and the coefficients, we will follow these steps:

### Step 1: Set the Equation to Zero
We start with the polynomial:
\[ x^2 - 36 = 0 \]

### Step 2: Solve for \(x\)
To find the zeroes, we rearrange the equation:
\[ x^2 = 36 \]
Next, we take the square root of both sides:
\[ x = \pm \sqrt{36} \]
This gives us:
\[ x = \pm 6 \]
Thus, the zeroes of the polynomial are:
\[ x = 6 \quad \text{and} \quad x = -6 \]

### Step 3: Calculate the Sum of the Zeroes
The sum of the zeroes can be calculated as:
\[ 6 + (-6) = 0 \]

### Step 4: Calculate the Product of the Zeroes
The product of the zeroes is:
\[ 6 \times (-6) = -36 \]

### Step 5: Identify Coefficients
Now, we identify the coefficients of the polynomial \(x^2 - 36\):
- The coefficient \(a\) (of \(x^2\)) is \(1\)
- The coefficient \(b\) (of \(x\)) is \(0\)
- The coefficient \(c\) (constant term) is \(-36\)

### Step 6: Verify the Relationships
According to the relationships for the sum and product of the zeroes:
- The sum of the zeroes is given by \(-\frac{b}{a}\):
\[ -\frac{0}{1} = 0 \]
- The product of the zeroes is given by \(\frac{c}{a}\):
\[ \frac{-36}{1} = -36 \]

### Conclusion
We found that:
- The sum of the zeroes \(= 0\) (matches with \(-\frac{b}{a}\))
- The product of the zeroes \(= -36\) (matches with \(\frac{c}{a}\))

Thus, we have verified the relationships between the zeroes and the coefficients.

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