Algebraic Manipulation: Higher - Go Teach Maths: Handcrafted ... - Free Printable

Let’s solve each problem one by one, step by step.

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① Simplify: 2c + c × 2c − 5c

First, remember order of operations (BIDMAS/PEMDAS): multiplication before addition/subtraction.

So, do c × 2c first → that’s 2c²

Now the expression becomes:
2c + 2c² − 5c

Now combine like terms:
2c − 5c = -3c

So final answer:
2c² − 3c

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② Expand: (8y³)²

This means square everything inside the brackets.

(8y³)² = 8² × (y³)² = 64 × y⁶ = 64y⁶

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③ Factorise: 12a² − 15a

Find the highest common factor (HCF) of 12 and 15 → that’s 3.
Both terms have at least one ‘a’ → so HCF is 3a

Factor out 3a:
12a² ÷ 3a = 4a
−15a ÷ 3a = −5

So: 3a(4a − 5)

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⑤ Make f the subject: h = 2f + g

We want to get f alone on one side.

Step 1: Subtract g from both sides:
h − g = 2f

Step 2: Divide both sides by 2:
f = (h − g)/2

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⑥ Make x the subject: y = √[(x + 1)/3]

Step 1: Square both sides to remove the square root:
y² = (x + 1)/3

Step 2: Multiply both sides by 3:
3y² = x + 1

Step 3: Subtract 1 from both sides:
x = 3y² − 1

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⑦ Expand & simplify: (x + 2)(x − 4)

Use FOIL (First, Outer, Inner, Last):

First: x × x = x²
Outer: x × (−4) = −4x
Inner: 2 × x = 2x
Last: 2 × (−4) = −8

Add them up:
x² − 4x + 2x − 8 = x² − 2x − 8

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⑨ Factorise: 2x² − 14x + 24

First, factor out the HCF if possible. All coefficients are even → factor out 2:

2(x² − 7x + 12)

Now factor the quadratic inside:
Need two numbers that multiply to 12 and add to −7 → those are −3 and −4

So:
2(x − 3)(x − 4)

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⑩ Factorise: x² − 4

This is a difference of squares: a² − b² = (a − b)(a + b)

Here, x² − 4 = x² − 2² → so:
(x − 2)(x + 2)

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⑪ Expand & simplify: (3x + 2)(x − 1)

FOIL again:

First: 3x × x = 3x²
Outer: 3x × (−1) = −3x
Inner: 2 × x = 2x
Last: 2 × (−1) = −2

Add them:
3x² − 3x + 2x − 2 = 3x² − x − 2

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⑬ Write in the form (x + a)² + b

Wait — the original problem for #13 is cut off in the image. But since it says “Write in the form (x + a)² + b”, this usually means completing the square.

But without the actual expression, we can’t solve it. Let me check the image again...

Actually, looking back — the user only provided part of the grid. Problems ④, , ⑫, are partially visible or cut off. Since they’re incomplete, I’ll skip them unless you provide full text.

Similarly, #14, #15, #16 are also cut off.

But let’s look at what’s fully visible:

#14: Simplify. x + 2 ÷ (2x / ?) — seems incomplete.

Actually, re-examining the image description — perhaps the user intended only the clearly visible problems? Or maybe there was a formatting issue.

Since the instruction says “solve the problem accurately” and the image has multiple questions, but some are cut off, I will solve all that are fully legible.

From above, we’ve done: ①, ②, ③, ⑤, ⑥, ⑦, ⑨, ⑩, ⑪

What about #13? It says “Write in the form (x + a)² + b” — but no expression is given. So I cannot solve it.

Same with others.

Perhaps the user meant to upload a complete sheet, but due to cropping, only parts are visible.

Given that, I’ll proceed with the ones I can solve completely.

Wait — actually, looking again at the original message, the user pasted an image description with 16 boxes, but many are truncated. For example:

Box ④: “Exp 2(x ...” — incomplete
Box ⑧: “Fac...” — incomplete
Box ⑫: “Exp...” — incomplete
Box ⑬: “Write in the form (x + a)^2 + b” — but no expression!
Box ⑭: “Simplify. x + 2 ÷ 2x” — ambiguous
Box ⑮: “Simplify. x_ + 4 _ 2” — unclear
Box ⑯: “Sim...” — cut off

So realistically, only boxes ①, ②, ③, ⑤, ⑥, ⑦, ⑨, ⑩, ⑪ are fully solvable.

I’ll list their answers now.

Final Answer:
① 2c² − 3c
② 64y⁶
③ 3a(4a − 5)
⑤ f = (h − g)/2
⑥ x = 3y² − 1
⑦ x² − 2x − 8
⑨ 2(x − 3)(x − 4)
⑩ (x − 2)(x + 2)
⑪ 3x² − x − 2