Sure, let me walk you through the detailed solutions to each expression step-by-step.
### Question 11: Expand the expression [tex]\((3x - 1)^2 - 6x + 2\)[/tex]
1. First, expand the squared term: [tex]\((3x - 1)^2\)[/tex]
- [tex]\((3x - 1)^2 = (3x - 1)(3x - 1)\)[/tex]
- Apply the distributive property:
[tex]\[
(3x - 1)(3x - 1) = 3x(3x) + 3x(-1) + (-1)(3x) + (-1)(-1)
\][/tex]
- Simplify the products:
[tex]\[
3x \cdot 3x = 9x^2, \quad 3x \cdot (-1) = -3x, \quad (-1) \cdot 3x = -3x, \quad (-1) \cdot (-1) = 1
\][/tex]
- Combine the terms:
[tex]\[
9x^2 - 3x - 3x + 1 = 9x^2 - 6x + 1
\][/tex]
2. Next, substitute this result back into the expression:
[tex]\[
(3x - 1)^2 - 6x + 2 = 9x^2 - 6x + 1 - 6x + 2
\][/tex]
3. Combine like terms:
[tex]\[
9x^2 - 6x - 6x + 1 + 2 = 9x^2 - 12x + 3
\][/tex]
So, the expanded expression is:
[tex]\[
9x^2 - 12x + 3
\][/tex]
### Question 13: Simplify the expression [tex]\(m(6 - m) - (6 - m)\)[/tex]
1. Distribute [tex]\(m\)[/tex] in the first term [tex]\(m(6 - m)\)[/tex]:
[tex]\[
m(6 - m) = m \cdot 6 - m \cdot m = 6m - m^2
\][/tex]
2. Now the expression becomes:
[tex]\[
6m - m^2 - (6 - m)
\][/tex]
3. Distribute the minus sign across [tex]\((6 - m)\)[/tex]:
[tex]\[
6m - m^2 - 6 + m
\][/tex]
4. Combine like terms:
[tex]\[
-m^2 + 6m + m - 6 = -m^2 + 7m - 6
\][/tex]
So, the simplified expression is:
[tex]\[
-m^2 + 7m - 6
\][/tex]
### Question 15: Simplify the expression [tex]\(a(a - 2b) + c(a - 2b) + (2b - a)\)[/tex]
1. Distribute [tex]\(a\)[/tex] in the first term [tex]\(a(a - 2b)\)[/tex]:
[tex]\[
a(a - 2b) = a^2 - 2ab
\][/tex]
2. Distribute [tex]\(c\)[/tex] in the second term [tex]\(c(a - 2b)\)[/tex]:
[tex]\[
c(a - 2b) = ca - 2cb
\][/tex]
3. Now, combine all the terms:
[tex]\[
a^2 - 2ab + ca - 2cb + (2b - a)
\][/tex]
4. Distribute the minus sign in the last term [tex]\((2b - a)\)[/tex]:
[tex]\[
a^2 - 2ab + ca - 2cb + 2b - a
\][/tex]
5. Combine like terms:
- Combine terms involving [tex]\(a\)[/tex]: [tex]\(a^2 - a\)[/tex]
- Combine terms involving [tex]\(b\)[/tex]: [tex]\(-2ab + 2b - 2cb\)[/tex]
So, the simplified expression is:
[tex]\[
a(a - 2b) - a + 2b + c(a - 2b)
\][/tex]
