https://file.aiquickdraw.com/mathbot/file/1726118745293wjsp3vkx.jpg

Let's go through each part of the question:

### (a) (i) Factorize \(4x^2 - 20xy + 25y^2\):

This expression is a quadratic trinomial. We need to factor it by treating it as a perfect square trinomial.

- Notice that the first term, \(4x^2\), is a perfect square: \((2x)^2\).
- The last term, \(25y^2\), is also a perfect square: \((5y)^2\).
- The middle term, \(-20xy\), should be double the product of the square roots of the first and last terms: \(2(2x)(5y) = 20xy\), but since the middle term has a minus sign, it fits.

Therefore, the factorization of \(4x^2 - 20xy + 25y^2\) is:
\[
(2x - 5y)^2
\]

### (a) (ii) Factorize \(4x^2 - 20xy + 25y^2 - 12x + 30y\) completely:

From part (i), we know that \(4x^2 - 20xy + 25y^2 = (2x - 5y)^2\).

So, we can rewrite the expression as:
\[
(2x - 5y)^2 - 12x + 30y
\]

Now, let's group the terms involving \(x\) and \(y\):
\[
(2x - 5y)^2 - 6(2x - 5y)
\]
Factor out the common factor \(2x - 5y\):
\[
(2x - 5y)((2x - 5y) - 6)
\]
Simplifying the second factor:
\[
(2x - 5y)(2x - 5y - 6) = (2x - 5y)(2x - 5y - 6)
\]

Thus, the complete factorization is:
\[
(2x - 5y)(2x - 5y - 6)
\]

### (b) (i) Evaluate \(k\) when \(x = -2\) and \(y = 1\):

The given equation is:
\[
\frac{x + ky}{2x - y} = \frac{1}{4}
\]

Substitute \(x = -2\) and \(y = 1\) into the equation:
\[
\frac{-2 + k(1)}{2(-2) - 1} = \frac{1}{4}
\]
This simplifies to:
\[
\frac{-2 + k}{-4 - 1} = \frac{1}{4}
\]
\[
\frac{-2 + k}{-5} = \frac{1}{4}
\]
Cross-multiply:
\[
4(-2 + k) = -5
\]
Simplify:
\[
-8 + 4k = -5
\]
Solve for \(k\):
\[
4k = -5 + 8
\]
\[
4k = 3
\]
\[
k = \frac{3}{4}
\]

### (b) (ii) Express \(y\) in terms of \(x\) and \(k\):

We start with the equation:
\[
\frac{x + ky}{2x - y} = \frac{1}{4}
\]
Cross-multiply:
\[
4(x + ky) = 2x - y
\]
Expand the left-hand side:
\[
4x + 4ky = 2x - y
\]
Rearrange the equation to group terms involving \(y\) on one side:
\[
4ky + y = 2x - 4x
\]
\[
y(4k + 1) = -2x
\]
Solve for \(y\):
\[
y = \frac{-2x}{4k + 1}
\]

---

Would you like further explanation on any of these steps?

Here are 5 related questions for practice:
1. Factorize \(9x^2 - 24xy + 16y^2\).
2. Solve \(3x + 2y = 7\) for \(y\).
3. Solve for \(k\) in \(\frac{kx + 3}{5} = 2\).
4. What is the factorization of \(x^2 + 6x + 9\)?
5. Simplify \(\frac{3x + 6}{x^2 - 9}\).

**Tip:** Always check if the quadratic expression can be factored as a perfect square trinomial before applying other methods.

Factorize the expressions and evaluate the given equation, including finding k and expressing y in terms of x and k.

Learn how to factor quadratic expressions like 4x^2 - 20xy + 25y^2 and solve linear equations involving variables k, x, and y. This solution includes step-by-step factorization and methods to express y in terms of x and k. Suitable for Grades 9-11.

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