Factoring Quadratics (D) worksheet featuring 20 quadratic expressions to factor.

To solve the problem of factoring each quadratic expression, we will use various techniques such as:

1. Factoring by Grouping
2. Difference of Squares
3. Perfect Square Trinomials
4. Trial and Error for General Quadratics

Let's go through each expression step by step.

---

Problem 1: \( 16x^2 - 16x + 3 \)

We look for two numbers that multiply to \( 16 \times 3 = 48 \) and add to \(-16\). These numbers are \(-12\) and \(-4\).

\[
16x^2 - 16x + 3 = 16x^2 - 12x - 4x + 3
\]

Group the terms:

\[
= (16x^2 - 12x) + (-4x + 3)
\]

Factor out the greatest common factor (GCF) from each group:

\[
= 4x(4x - 3) - 1(4x - 3)
\]

Factor out the common binomial factor \( (4x - 3) \):

\[
= (4x - 3)(4x - 1)
\]

Answer:
\[
\boxed{(4x - 3)(4x - 1)}
\]

---

Problem 2: \( 12x^2 - x - 20 \)

We look for two numbers that multiply to \( 12 \times -20 = -240 \) and add to \(-1\). These numbers are \(-16\) and \(15\).

\[
12x^2 - x - 20 = 12x^2 - 16x + 15x - 20
\]

Group the terms:

\[
= (12x^2 - 16x) + (15x - 20)
\]

Factor out the GCF from each group:

\[
= 4x(3x - 4) + 5(3x - 4)
\]

Factor out the common binomial factor \( (3x - 4) \):

\[
= (3x - 4)(4x + 5)
\]

Answer:
\[
\boxed{(3x - 4)(4x + 5)}
\]

---

Problem 3: \( 42x^2 - 75x + 27 \)

First, find the GCF of the coefficients \( 42, 75, \) and \( 27 \), which is \( 3 \). Factor out \( 3 \):

\[
42x^2 - 75x + 27 = 3(14x^2 - 25x + 9)
\]

Now, factor \( 14x^2 - 25x + 9 \). We look for two numbers that multiply to \( 14 \times 9 = 126 \) and add to \(-25\). These numbers are \(-18\) and \(-7\).

\[
14x^2 - 25x + 9 = 14x^2 - 18x - 7x + 9
\]

Group the terms:

\[
= (14x^2 - 18x) + (-7x + 9)
\]

Factor out the GCF from each group:

\[
= 2x(7x - 9) - 1(7x - 9)
\]

Factor out the common binomial factor \( (7x - 9) \):

\[
= (7x - 9)(2x - 1)
\]

So, the complete factorization is:

\[
42x^2 - 75x + 27 = 3(7x - 9)(2x - 1)
\]

Answer:
\[
\boxed{3(7x - 9)(2x - 1)}
\]

---

Problem 4: \( 72x^2 + 2x - 4 \)

First, find the GCF of the coefficients \( 72, 2, \) and \( -4 \), which is \( 2 \). Factor out \( 2 \):

\[
72x^2 + 2x - 4 = 2(36x^2 + x - 2)
\]

Now, factor \( 36x^2 + x - 2 \). We look for two numbers that multiply to \( 36 \times -2 = -72 \) and add to \( 1 \). These numbers are \( 9 \) and \(-8\).

\[
36x^2 + x - 2 = 36x^2 + 9x - 8x - 2
\]

Group the terms:

\[
= (36x^2 + 9x) + (-8x - 2)
\]

Factor out the GCF from each group:

\[
= 9x(4x + 1) - 2(4x + 1)
\]

Factor out the common binomial factor \( (4x + 1) \):

\[
= (4x + 1)(9x - 2)
\]

So, the complete factorization is:

\[
72x^2 + 2x - 4 = 2(4x + 1)(9x - 2)
\]

Answer:
\[
\boxed{2(4x + 1)(9x - 2)}
\]

---

Problem 5: \( 8x^2 - 22x - 63 \)

We look for two numbers that multiply to \( 8 \times -63 = -504 \) and add to \(-22\). These numbers are \(-36\) and \(14\).

\[
8x^2 - 22x - 63 = 8x^2 - 36x + 14x - 63
\]

Group the terms:

\[
= (8x^2 - 36x) + (14x - 63)
\]

Factor out the GCF from each group:

\[
= 4x(2x - 9) + 7(2x - 9)
\]

Factor out the common binomial factor \( (2x - 9) \):

\[
= (2x - 9)(4x + 7)
\]

Answer:
\[
\boxed{(2x - 9)(4x + 7)}
\]

---

Problem 6: \( 15x^2 + 20x + 5 \)

First, find the GCF of the coefficients \( 15, 20, \) and \( 5 \), which is \( 5 \). Factor out \( 5 \):

\[
15x^2 + 20x + 5 = 5(3x^2 + 4x + 1)
\]

Now, factor \( 3x^2 + 4x + 1 \). We look for two numbers that multiply to \( 3 \times 1 = 3 \) and add to \( 4 \). These numbers are \( 3 \) and \( 1 \).

\[
3x^2 + 4x + 1 = 3x^2 + 3x + x + 1
\]

Group the terms:

\[
= (3x^2 + 3x) + (x + 1)
\]

Factor out the GCF from each group:

\[
= 3x(x + 1) + 1(x + 1)
\]

Factor out the common binomial factor \( (x + 1) \):

\[
= (x + 1)(3x + 1)
\]

So, the complete factorization is:

\[
15x^2 + 20x + 5 = 5(x + 1)(3x + 1)
\]

Answer:
\[
\boxed{5(x + 1)(3x + 1)}
\]

---

Problem 7: \( 24x^2 + 72x + 54 \)

First, find the GCF of the coefficients \( 24, 72, \) and \( 54 \), which is \( 6 \). Factor out \( 6 \):

\[
24x^2 + 72x + 54 = 6(4x^2 + 12x + 9)
\]

Now, factor \( 4x^2 + 12x + 9 \). Notice that this is a perfect square trinomial:

\[
4x^2 + 12x + 9 = (2x + 3)^2
\]

So, the complete factorization is:

\[
24x^2 + 72x + 54 = 6(2x + 3)^2
\]

Answer:
\[
\boxed{6(2x + 3)^2}
\]

---

Problem 8: \( x^2 - 11x + 30 \)

We look for two numbers that multiply to \( 1 \times 30 = 30 \) and add to \(-11\). These numbers are \(-6\) and \(-5\).

\[
x^2 - 11x + 30 = (x - 6)(x - 5)
\]

Answer:
\[
\boxed{(x - 6)(x - 5)}
\]

---

Problem 9: \( 6x^2 + 40x - 14 \)

First, find the GCF of the coefficients \( 6, 40, \) and \( -14 \), which is \( 2 \). Factor out \( 2 \):

\[
6x^2 + 40x - 14 = 2(3x^2 + 20x - 7)
\]

Now, factor \( 3x^2 + 20x - 7 \). We look for two numbers that multiply to \( 3 \times -7 = -21 \) and add to \( 20 \). These numbers are \( 21 \) and \(-1\).

\[
3x^2 + 20x - 7 = 3x^2 + 21x - x - 7
\]

Group the terms:

\[
= (3x^2 + 21x) + (-x - 7)
\]

Factor out the GCF from each group:

\[
= 3x(x + 7) - 1(x + 7)
\]

Factor out the common binomial factor \( (x + 7) \):

\[
= (x + 7)(3x - 1)
\]

So, the complete factorization is:

\[
6x^2 + 40x - 14 = 2(x + 7)(3x - 1)
\]

Answer:
\[
\boxed{2(x + 7)(3x - 1)}
\]

---

Problem 10: \( 56x^2 - 24x - 32 \)

First, find the GCF of the coefficients \( 56, -24, \) and \(-32\), which is \( 8 \). Factor out \( 8 \):

\[
56x^2 - 24x - 32 = 8(7x^2 - 3x - 4)
\]

Now, factor \( 7x^2 - 3x - 4 \). We look for two numbers that multiply to \( 7 \times -4 = -28 \) and add to \(-3\). These numbers are \(-7\) and \(4\).

\[
7x^2 - 3x - 4 = 7x^2 - 7x + 4x - 4
\]

Group the terms:

\[
= (7x^2 - 7x) + (4x - 4)
\]

Factor out the GCF from each group:

\[
= 7x(x - 1) + 4(x - 1)
\]

Factor out the common binomial factor \( (x - 1) \):

\[
= (x - 1)(7x + 4)
\]

So, the complete factorization is:

\[
56x^2 - 24x - 32 = 8(x - 1)(7x + 4)
\]

Answer:
\[
\boxed{8(x - 1)(7x + 4)}
\]

---

Final Answer:

\[
\boxed{
\begin{aligned}
1. & \ (4x - 3)(4x - 1) \\
2. & \ (3x - 4)(4x + 5) \\
3. & \ 3(7x - 9)(2x - 1) \\
4. & \ 2(4x + 1)(9x - 2) \\
5. & \ (2x - 9)(4x + 7) \\
6. & \ 5(x + 1)(3x + 1) \\
7. & \ 6(2x + 3)^2 \\
8. & \ (x - 6)(x - 5) \\
9. & \ 2(x + 7)(3x - 1) \\
10. & \ 8(x - 1)(7x + 4) \\
\end{aligned}
}
\]