(i) For a quadratic polynomial with equal zeroes, the graph will touch the x-axis at a single point. This means the discriminant (\(b^2 - 4ac\)) is equal to zero, indicating that both zeroes are the same. A rough figure might look like this:
```
x
|
|
|
|
| *
| / \
| / \
| / \
+----------------------> y
```
(ii) For a quadratic polynomial with unequal zeroes, the graph will intersect the x-axis at two distinct points. This means the discriminant (\(b^2 - 4ac\)) is greater than zero, indicating that both zeroes are different. A rough figure might look like this:
```
x
|
|
|
|
| *
| *
| *
| *
+----------------------> y
```
In both figures, the points marked with '*' represent the zeroes of the quadratic polynomial.
Answers & Comments
Answer:
(i) For a quadratic polynomial with equal zeroes, the graph will touch the x-axis at a single point. This means the discriminant (\(b^2 - 4ac\)) is equal to zero, indicating that both zeroes are the same. A rough figure might look like this:
```
x
|
|
|
|
| *
| / \
| / \
| / \
+----------------------> y
```
(ii) For a quadratic polynomial with unequal zeroes, the graph will intersect the x-axis at two distinct points. This means the discriminant (\(b^2 - 4ac\)) is greater than zero, indicating that both zeroes are different. A rough figure might look like this:
```
x
|
|
|
|
| *
| *
| *
| *
+----------------------> y
```
In both figures, the points marked with '*' represent the zeroes of the quadratic polynomial.