https://file.aiquickdraw.com/mathbot/file/17273447586876joe40jk.jpg

Let's solve the question based on the diagram and the given information.

### Part (a)(i): Show that \( 2x^2 + 11x - 105 = 0 \)

The diagram shows a right-angled triangle \( \triangle ABC \), where:

- \( AC = x + 3 \) cm
- \( AB = 2x + 5 \) cm
- The area of the triangle is 60 cm².

The area of a triangle is given by the formula:

\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]

Here, \( AB \) is the base and \( AC \) is the height.

So, applying the formula for area:

\[
60 = \frac{1}{2} \times (2x + 5) \times (x + 3)
\]

Now, multiply both sides by 2 to eliminate the fraction:

\[
120 = (2x + 5)(x + 3)
\]

Expand the right-hand side:

\[
120 = 2x(x + 3) + 5(x + 3)
\]
\[
120 = 2x^2 + 6x + 5x + 15
\]
\[
120 = 2x^2 + 11x + 15
\]

Now, subtract 120 from both sides:

\[
0 = 2x^2 + 11x + 15 - 120
\]
\[
0 = 2x^2 + 11x - 105
\]

This matches the given equation \( 2x^2 + 11x - 105 = 0 \).

---

### Part (a)(ii): Solve by Factorisation

We need to solve the quadratic equation:

\[
2x^2 + 11x - 105 = 0
\]

To factor this, first find two numbers that multiply to \( 2 \times (-105) = -210 \) and add to 11. These numbers are 21 and -10.

Now, rewrite the middle term using these values:

\[
2x^2 + 21x - 10x - 105 = 0
\]

Group the terms:

\[
(2x^2 + 21x) - (10x + 105) = 0
\]

Factor each group:

\[
x(2x + 21) - 5(2x + 21) = 0
\]

Now factor out \( (2x + 21) \):

\[
(2x + 21)(x - 5) = 0
\]

Set each factor equal to zero:

1. \( 2x + 21 = 0 \) gives \( x = -\frac{21}{2} = -10.5 \)
2. \( x - 5 = 0 \) gives \( x = 5 \)

Thus, the solutions are:

\[
x = 5 \quad \text{or} \quad x = -10.5
\]

Since \( x \) represents a length in this context, only the positive solution \( x = 5 \) is valid.

---

Would you like further details or explanations on any of these steps?

---

### Follow-up Questions:

1. How do you solve quadratic equations using other methods like completing the square?
2. How do you verify if your factorization is correct?
3. What is the significance of rejecting the negative solution in this context?
4. Can you solve this equation using the quadratic formula instead of factorization?
5. How does changing the area of the triangle affect the quadratic equation?

### Tip: 
Always check your factorizations by expanding the factors to ensure they give back the original quadratic equation.

The diagram shows a right-angled triangle ABC. Given the area of the triangle is 60 cm², show that 2x² + 11x - 105 = 0 and solve this equation by factorization.

This problem involves solving a quadratic equation derived from the area of a right-angled triangle. Learn how to show that 2x² + 11x - 105 = 0 and solve it by factorization. Perfect for students in Grades 9-11.

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