Question
Factorise completely
1. 5 a squared + 27 a + 10
2. 24 x squared - 30 x + 8 + 2 x
3. 3 x squared - 5 x - 12
4. 15 y squared - 11 y + 2
5. 12 x squared - 16 x + 4 + 2 x
6. 5 a squared + 27 a + 10 + 4 a squared - b squared
7. 2 x squared - 18 + 5 a to the power of 4 - 20 b squared
1. 5 a squared + 27 a + 10
2. 24 x squared - 30 x + 8 + 2 x
3. 3 x squared - 5 x - 12
4. 15 y squared - 11 y + 2
5. 12 x squared - 16 x + 4 + 2 x
6. 5 a squared + 27 a + 10 + 4 a squared - b squared
7. 2 x squared - 18 + 5 a to the power of 4 - 20 b squared
Answers
GPT-4o mini
Let's factor each expression step by step.
### 1. \(5a^2 + 27a + 10\)
To factor, we look for two numbers that multiply to \(5 \times 10 = 50\) and add to \(27\). The appropriate numbers are \(25\) and \(2\).
Rewriting the middle term:
\[
5a^2 + 25a + 2a + 10
\]
Grouping:
\[
(5a^2 + 25a) + (2a + 10) = 5a(a + 5) + 2(a + 5) = (5a + 2)(a + 5)
\]
### 2. \(24x^2 - 30x + 8 + 2x\)
First, simplify the expression:
\[
24x^2 - 30x + 8 + 2x = 24x^2 - 28x + 8
\]
Now, look for two numbers that multiply to \(24 \times 8 = 192\) and add to \(-28\). The appropriate numbers are \(-24\) and \(-4\).
Rewriting the middle term:
\[
24x^2 - 24x - 4x + 8
\]
Grouping:
\[
(24x^2 - 24x) + (-4x + 8) = 24x(x - 1) - 4(x - 2) = 4(6x - 2)(x - 1)
\]
We can further simplify:
\[
4(2(3x - 1))(x - 1) = 8(3x - 1)(x - 1)
\]
### 3. \(3x^2 - 5x - 12\)
We need two numbers that multiply to \(3 \times (-12) = -36\) and add to \(-5\). The numbers are \(-9\) and \(4\).
Rewriting:
\[
3x^2 - 9x + 4x - 12
\]
Grouping:
\[
(3x^2 - 9x) + (4x - 12) = 3x(x - 3) + 4(x - 3) = (3x + 4)(x - 3)
\]
### 4. \(15y^2 - 11y + 2\)
We need numbers that multiply to \(15 \times 2 = 30\) and add to \(-11\). The numbers are \(-6\) and \(-5\).
Rewriting:
\[
15y^2 - 6y - 5y + 2
\]
Grouping:
\[
(15y^2 - 6y) + (-5y + 2) = 3y(5y - 2) - 1(5y - 2) = (5y - 2)(3y - 1)
\]
### 5. \(12x^2 - 16x + 4 + 2x\)
First, simplify:
\[
12x^2 - 16x + 4 + 2x = 12x^2 - 14x + 4
\]
We need two numbers that multiply to \(12 \times 4 = 48\) and add to \(-14\). The numbers are \(-6\) and \(-8\).
Rewriting:
\[
12x^2 - 6x - 8x + 4
\]
Grouping:
\[
(12x^2 - 6x) + (-8x + 4) = 6x(2x - 1) - 4(2x - 1) = (2x - 1)(6x - 4)
\]
We can factor out \(2\) from \(6x - 4\):
\[
(2x - 1)(2(3x - 2)) = 2(2x - 1)(3x - 2)
\]
### 6. \(5a^2 + 27a + 10 + 4a^2 - b^2\)
Combine like terms:
\[
(5a^2 + 4a^2) + 27a + 10 - b^2 = 9a^2 + 27a + 10 - b^2
\]
Factor this expression based on the quadratic portion:
Looking for factors for \(9a^2 + 27a + 10\) (similar method as above):
Multiplying \(9 \times 10 = 90\) with sums to \(27\):
Rewriting:
\[
(9a^2 + 18a + 9a + 10) = 9a(a + 2) + 5(a + 2) = (9a + 5)(a + 2) - b^2
\]
This can also be recognized as a difference of squares:
\[
(9a + 5)(a + 2) - (b)(b) = [(9a + 5) + b][(9a + 5) - b]
\]
### 7. \(2x^2 - 18 + 5a^4 - 20b^2\)
First, group the terms:
\[
2x^2 + 5a^4 - 20b^2 - 18 = 2x^2 + 5(a^4 - 4b^2) - 18
\]
Factor \(a^4 - 4b^2\) (a difference of squares):
\[
a^4 - 4b^2 = (a^2 - 2b)(a^2 + 2b)
\]
Thus,
\[
2x^2 - 18 + 5[(a^2 - 2b)(a^2 + 2b)] = 2(x^2 - 9) + 5(a^2 - 2b)(a^2 + 2b)
\]
This can also be expressed as \((x - 3)(x + 3)\):
Combining:
\[
2(x - 3)(x + 3) + 5(a^2 - 2b)(a^2 + 2b)
\]
We observe the complete factorization involves the individual components.
### Summary of Factored Forms:
1. \((5a + 2)(a + 5)\)
2. \(8(3x - 1)(x - 1)\)
3. \((3x + 4)(x - 3)\)
4. \((5y - 2)(3y - 1)\)
5. \(2(2x - 1)(3x - 2)\)
6. \([(9a + 5) + b][(9a + 5) - b]\)
7. \(2(x - 3)(x + 3) + 5(a^2 - 2b)(a^2 + 2b)\)
### 1. \(5a^2 + 27a + 10\)
To factor, we look for two numbers that multiply to \(5 \times 10 = 50\) and add to \(27\). The appropriate numbers are \(25\) and \(2\).
Rewriting the middle term:
\[
5a^2 + 25a + 2a + 10
\]
Grouping:
\[
(5a^2 + 25a) + (2a + 10) = 5a(a + 5) + 2(a + 5) = (5a + 2)(a + 5)
\]
### 2. \(24x^2 - 30x + 8 + 2x\)
First, simplify the expression:
\[
24x^2 - 30x + 8 + 2x = 24x^2 - 28x + 8
\]
Now, look for two numbers that multiply to \(24 \times 8 = 192\) and add to \(-28\). The appropriate numbers are \(-24\) and \(-4\).
Rewriting the middle term:
\[
24x^2 - 24x - 4x + 8
\]
Grouping:
\[
(24x^2 - 24x) + (-4x + 8) = 24x(x - 1) - 4(x - 2) = 4(6x - 2)(x - 1)
\]
We can further simplify:
\[
4(2(3x - 1))(x - 1) = 8(3x - 1)(x - 1)
\]
### 3. \(3x^2 - 5x - 12\)
We need two numbers that multiply to \(3 \times (-12) = -36\) and add to \(-5\). The numbers are \(-9\) and \(4\).
Rewriting:
\[
3x^2 - 9x + 4x - 12
\]
Grouping:
\[
(3x^2 - 9x) + (4x - 12) = 3x(x - 3) + 4(x - 3) = (3x + 4)(x - 3)
\]
### 4. \(15y^2 - 11y + 2\)
We need numbers that multiply to \(15 \times 2 = 30\) and add to \(-11\). The numbers are \(-6\) and \(-5\).
Rewriting:
\[
15y^2 - 6y - 5y + 2
\]
Grouping:
\[
(15y^2 - 6y) + (-5y + 2) = 3y(5y - 2) - 1(5y - 2) = (5y - 2)(3y - 1)
\]
### 5. \(12x^2 - 16x + 4 + 2x\)
First, simplify:
\[
12x^2 - 16x + 4 + 2x = 12x^2 - 14x + 4
\]
We need two numbers that multiply to \(12 \times 4 = 48\) and add to \(-14\). The numbers are \(-6\) and \(-8\).
Rewriting:
\[
12x^2 - 6x - 8x + 4
\]
Grouping:
\[
(12x^2 - 6x) + (-8x + 4) = 6x(2x - 1) - 4(2x - 1) = (2x - 1)(6x - 4)
\]
We can factor out \(2\) from \(6x - 4\):
\[
(2x - 1)(2(3x - 2)) = 2(2x - 1)(3x - 2)
\]
### 6. \(5a^2 + 27a + 10 + 4a^2 - b^2\)
Combine like terms:
\[
(5a^2 + 4a^2) + 27a + 10 - b^2 = 9a^2 + 27a + 10 - b^2
\]
Factor this expression based on the quadratic portion:
Looking for factors for \(9a^2 + 27a + 10\) (similar method as above):
Multiplying \(9 \times 10 = 90\) with sums to \(27\):
Rewriting:
\[
(9a^2 + 18a + 9a + 10) = 9a(a + 2) + 5(a + 2) = (9a + 5)(a + 2) - b^2
\]
This can also be recognized as a difference of squares:
\[
(9a + 5)(a + 2) - (b)(b) = [(9a + 5) + b][(9a + 5) - b]
\]
### 7. \(2x^2 - 18 + 5a^4 - 20b^2\)
First, group the terms:
\[
2x^2 + 5a^4 - 20b^2 - 18 = 2x^2 + 5(a^4 - 4b^2) - 18
\]
Factor \(a^4 - 4b^2\) (a difference of squares):
\[
a^4 - 4b^2 = (a^2 - 2b)(a^2 + 2b)
\]
Thus,
\[
2x^2 - 18 + 5[(a^2 - 2b)(a^2 + 2b)] = 2(x^2 - 9) + 5(a^2 - 2b)(a^2 + 2b)
\]
This can also be expressed as \((x - 3)(x + 3)\):
Combining:
\[
2(x - 3)(x + 3) + 5(a^2 - 2b)(a^2 + 2b)
\]
We observe the complete factorization involves the individual components.
### Summary of Factored Forms:
1. \((5a + 2)(a + 5)\)
2. \(8(3x - 1)(x - 1)\)
3. \((3x + 4)(x - 3)\)
4. \((5y - 2)(3y - 1)\)
5. \(2(2x - 1)(3x - 2)\)
6. \([(9a + 5) + b][(9a + 5) - b]\)
7. \(2(x - 3)(x + 3) + 5(a^2 - 2b)(a^2 + 2b)\)