The image presents several quadratic expressions. We need to factorize the following expressions: 23. $4a^2 - 4a + 1$ 24. $12x^2 + 17x - 14$ 25. $48x^2 + 46x + 5$ 26. $64y^2 + 4y - 3$ 27. $9x^2 - 1$ 28. $4a^2 - 9$

1. Problem Description

The image presents several quadratic expressions. We need to factorize the following expressions:

2

3. $4a^2 - 4a + 1$

2

4. $12x^2 + 17x - 14$

2

5. $48x^2 + 46x + 5$

2

6. $64y^2 + 4y - 3$

2

7. $9x^2 - 1$

2

8. $4a^2 - 9$

2. Solution Steps

2

9. $4a^2 - 4a + 1$

This expression is a perfect square trinomial. We can rewrite it as $(2a)^2 - 2(2a)(1) + (1)^2$.

Using the formula $(x-y)^2 = x^2 - 2xy + y^2$, we can factor it as:

$4a^2 - 4a + 1 = (2a - 1)^2 = (2a - 1)(2a - 1)$.

3

0. $12x^2 + 17x - 14$

We are looking for two numbers that multiply to $(12)(-14) = -168$ and add up to $17$. The numbers are $24$ and $-7$.

Rewrite the middle term as $17x = 24x - 7x$.

$12x^2 + 17x - 14 = 12x^2 + 24x - 7x - 14$

Factor by grouping: $12x(x + 2) - 7(x + 2) = (12x - 7)(x + 2)$.

3

1. $48x^2 + 46x + 5$

We are looking for two numbers that multiply to $(48)(5) = 240$ and add up to $46$. The numbers are $6$ and $40$.

Rewrite the middle term as $46x = 6x + 40x$.

$48x^2 + 46x + 5 = 48x^2 + 6x + 40x + 5$

Factor by grouping: $6x(8x + 1) + 5(8x + 1) = (6x + 5)(8x + 1)$.

3

2. $64y^2 + 4y - 3$

We are looking for two numbers that multiply to $(64)(-3) = -192$ and add up to $4$. The numbers are $16$ and $-12$.

Rewrite the middle term as $4y = 16y - 12y$.

$64y^2 + 4y - 3 = 64y^2 + 16y - 12y - 3$

Factor by grouping: $16y(4y + 1) - 3(4y + 1) = (16y - 3)(4y + 1)$.

3

3. $9x^2 - 1$

This expression is a difference of squares. We can rewrite it as $(3x)^2 - (1)^2$.

Using the formula $a^2 - b^2 = (a - b)(a + b)$, we can factor it as:

$9x^2 - 1 = (3x - 1)(3x + 1)$.

3

4. $4a^2 - 9$

This expression is also a difference of squares. We can rewrite it as $(2a)^2 - (3)^2$.

Using the formula $a^2 - b^2 = (a - b)(a + b)$, we can factor it as:

$4a^2 - 9 = (2a - 3)(2a + 3)$.

3. Final Answer

3

5. $(2a - 1)(2a - 1)$

3

6. $(12x - 7)(x + 2)$

3

7. $(6x + 5)(8x + 1)$

3

8. $(16y - 3)(4y + 1)$

3

9. $(3x - 1)(3x + 1)$

4

0. $(2a - 3)(2a + 3)$