Algebra 1 Worksheets with Answers PDF | Printable Algebra 1 Math ... - Free Printable

Let’s solve each problem step by step. We’ll start with Section A, then Section B, and finally the Extension.

---

Section A: Simplify then factorise

We need to first simplify (expand and combine like terms), then factorise the quadratic expression.

---

1) x² – 6x – 2x + 12

Simplify:
x² – 8x + 12

Factorise: Find two numbers that multiply to 12 and add to -8 → -6 and -2
→ (x – 6)(x – 2)

✔ Final: (x – 6)(x – 2)

---

2) d(d – 5) – 84

Expand:
d² – 5d – 84

Factorise: Two numbers that multiply to -84 and add to -5 → -12 and 7
→ (d – 12)(d + 7)

✔ Final: (d – 12)(d + 7)

---

3) b² + 2(b – 4)

Expand:
b² + 2b – 8

Factorise: Multiply to -8, add to 2 → 4 and -2
→ (b + 4)(b – 2)

✔ Final: (b + 4)(b – 2)

---

4) x² – 3(2x + 9)

Expand:
x² – 6x – 27

Factorise: Multiply to -27, add to -6 → -9 and 3
→ (x – 9)(x + 3)

✔ Final: (x – 9)(x + 3)

---

5) c(c + 8) – 48

Expand:
c² + 8c – 48

Factorise: Multiply to -48, add to 8 → 12 and -4
→ (c + 12)(c – 4)

✔ Final: (c + 12)(c – 4)

---

6) 3a(a – 2) – 4a + 3

Expand:
3a² – 6a – 4a + 3 = 3a² – 10a + 3

Factorise: Use AC method or trial.
Multiply 3 × 3 = 9. Find factors of 9 that add to -10 → -9 and -1
Split middle term:
3a² – 9a – a + 3
Group: (3a² – 9a) + (-a + 3) = 3a(a – 3) –1(a – 3)
→ (3a – 1)(a – 3)

✔ Final: (3a – 1)(a – 3)

---

7) 5w(w – 2) – 4w – 3

Expand:
5w² – 10w – 4w – 3 = 5w² – 14w – 3

Factorise: 5 × -3 = -15. Factors of -15 that add to -14 → -15 and 1
Split: 5w² – 15w + w – 3
Group: 5w(w – 3) + 1(w – 3)
→ (5w + 1)(w – 3)

✔ Final: (5w + 1)(w – 3)

---

8) 3(6 – 5s) + s² + s²

First, combine s² + s² = 2s²
Then expand: 18 – 15s + 2s²
Rewrite in order: 2s² – 15s + 18

Factorise: 2 × 18 = 36. Factors of 36 that add to -15 → -12 and -3
Split: 2s² – 12s – 3s + 18
Group: 2s(s – 6) –3(s – 6)
→ (2s – 3)(s – 6)

✔ Final: (2s – 3)(s – 6)

---

9) 3 + 2y(4y + 5)

Expand:
3 + 8y² + 10y → 8y² + 10y + 3

Factorise: 8 × 3 = 24. Factors of 24 that add to 10 → 6 and 4
Split: 8y² + 6y + 4y + 3
Group: 2y(4y + 3) + 1(4y + 3)
→ (2y + 1)(4y + 3)

✔ Final: (2y + 1)(4y + 3)

---

10) 9x² – (x – 3)²

First, expand (x – 3)² = x² – 6x + 9
So: 9x² – (x² – 6x + 9) = 9x² – x² + 6x – 9 = 8x² + 6x – 9

Factorise: 8 × -9 = -72. Factors of -72 that add to 6 → 12 and -6
Split: 8x² + 12x – 6x – 9
Group: 4x(2x + 3) –3(2x + 3)
→ (4x – 3)(2x + 3)

✔ Final: (4x – 3)(2x + 3)

---

Section B: Factorise using difference of squares or other methods

Remember: Difference of squares = a² – b² = (a – b)(a + b)

Also look for common factors first!

---

1) x² – 4 → (x – 2)(x + 2)

2) s² – 25 → (s – 5)(s + 5)

3) t² – 64 → (t – 8)(t + 8)

4) 9 – y² → (3 – y)(3 + y) [or -(y – 3)(y + 3), but usually written as (3 – y)(3 + y)]

5) 49 – p² → (7 – p)(7 + p)

6) 4q² – 121 → (2q – 11)(2q + 11)

7) 81 – 25k² → (9 – 5k)(9 + 5k)

8) 1 – 400d² → (1 – 20d)(1 + 20d)

9) 600v² – 6 → First factor out 6: 6(100v² – 1) → 6(10v – 1)(10v + 1)

10) a² – b² → (a – b)(a + b)

11) x² – 9y² → (x – 3y)(x + 3y)

12) 4c² – d² → (2c – d)(2c + d)

13) 16s² – 9t² → (4s – 3t)(4s + 3t)

14) 49w² – 100v² → (7w – 10v)(7w + 10v)

15) 32p² – 18q² → Factor out 2: 2(16p² – 9q²) → 2(4p – 3q)(4p + 3q)

16) 48x² – 12y² → Factor out 12: 12(4x² – y²) → 12(2x – y)(2x + y)

17) 45a² – 125b² → Factor out 5: 5(9a² – 25b²) → 5(3a – 5b)(3a + 5b)

18) 72x² – 242y² → Factor out 2: 2(36x² – 121y²) → 2(6x – 11y)(6x + 11y)

19) a²b² – c² → (ab – c)(ab + c)

20) 9s – 4s³ → Rearrange: -4s³ + 9s → Factor out -s? Or better: factor out s first: s(9 – 4s²) → s(3 – 2s)(3 + 2s)

Wait — let’s write it as: 9s – 4s³ = s(9 – 4s²) = s(3 – 2s)(3 + 2s)

✔ Yes.

21) (xy)² – 4z² → (xy – 2z)(xy + 2z)

22) 64t⁴ – 16s⁴ → Factor out 16: 16(4t⁴ – s⁴) → Now 4t⁴ – s⁴ is difference of squares: (2t² – s²)(2t² + s²)
But 2t² – s² can be factored again if we want? Wait — no, unless specified. But actually, 4t⁴ – s⁴ = (2t²)² – (s²)² = (2t² – s²)(2t² + s²)
And 2t² – s² is not factorable over integers. So final: 16(2t² – s²)(2t² + s²)

BUT — wait! 64t⁴ – 16s⁴ = 16(4t⁴ – s⁴) = 16[(2t²)² – (s²)²] = 16(2t² – s²)(2t² + s²)

Alternatively, notice 64t⁴ – 16s⁴ = (8t²)² – (4s²)² → (8t² – 4s²)(8t² + 4s²) → then factor each: 4(2t² – s²) * 4(2t² + s²) = 16(2t² – s²)(2t² + s²) — same thing.

✔ Final: 16(2t² – s²)(2t² + s²)

23) (4x²)² – 36y² → 16x⁴ – 36y² → Factor out 4: 4(4x⁴ – 9y²) → Now 4x⁴ – 9y² = (2x²)² – (3y)² → (2x² – 3y)(2x² + 3y)
So overall: 4(2x² – 3y)(2x² + 3y)

✔ Final: 4(2x² – 3y)(2x² + 3y)

24) 27a⁴ – 12b² → Factor out 3: 3(9a⁴ – 4b²) → 9a⁴ – 4b² = (3a²)² – (2b)² → (3a² – 2b)(3a² + 2b)
So: 3(3a² – 2b)(3a² + 2b)

✔ Final: 3(3a² – 2b)(3a² + 2b)

---

Extension: Using difference of two squares

1) 4x² – (x – 2)²

This is already in form A² – B² where A = 2x, B = (x – 2)

So: [2x – (x – 2)][2x + (x – 2)]
= (2x – x + 2)(2x + x – 2)
= (x + 2)(3x – 2)

✔ Final: (x + 2)(3x – 2)

---

2) (2x + 1)² – (x – 4)²

Again, A² – B² → A = (2x + 1), B = (x – 4)

So: [(2x + 1) – (x – 4)][(2x + 1) + (x – 4)]
= (2x + 1 – x + 4)(2x + 1 + x – 4)
= (x + 5)(3x – 3)

Now factor further: 3x – 3 = 3(x – 1)
So: (x + 5) * 3(x – 1) = 3(x + 5)(x – 1)

✔ Final: 3(x + 5)(x – 1)

---

Final Answer:

Section A:
1) (x – 6)(x – 2)
2) (d – 12)(d + 7)
3) (b + 4)(b – 2)
4) (x – 9)(x + 3)
5) (c + 12)(c – 4)
6) (3a – 1)(a – 3)
7) (5w + 1)(w – 3)
8) (2s – 3)(s – 6)
9) (2y + 1)(4y + 3)
10) (4x – 3)(2x + 3)

Section B:
1) (x – 2)(x + 2)
2) (s – 5)(s + 5)
3) (t – 8)(t + 8)
4) (3 – y)(3 + y)
5) (7 – p)(7 + p)
6) (2q – 11)(2q + 11)
7) (9 – 5k)(9 + 5k)
8) (1 – 20d)(1 + 20d)
9) 6(10v – 1)(10v + 1)
10) (a – b)(a + b)
11) (x – 3y)(x + 3y)
12) (2c – d)(2c + d)
13) (4s – 3t)(4s + 3t)
14) (7w – 10v)(7w + 10v)
15) 2(4p – 3q)(4p + 3q)
16) 12(2x – y)(2x + y)
17) 5(3a – 5b)(3a + 5b)
18) 2(6x – 11y)(6x + 11y)
19) (ab – c)(ab + c)
20) s(3 – 2s)(3 + 2s)
21) (xy – 2z)(xy + 2z)
22) 16(2t² – s²)(2t² + s²)
23) 4(2x² – 3y)(2x² + 3y)
24) 3(3a² – 2b)(3a² + 2b)

Extension:
1) (x + 2)(3x – 2)
2) 3(x + 5)(x – 1)