Sure, let's solve the given expressions step-by-step.
### Equation 1: Solving for x
Given the equation:
[tex]\[ 4x + 2(2x - 5) = (x - 3) - (x - 8) \][/tex]
1. Distribute and expand both sides of the equation:
- Left-hand side (LHS): Distribute the 2 inside the parentheses.
[tex]\[
4x + 2(2x - 5) = 4x + 4x - 10
\][/tex]
- Right-hand side (RHS): Expand the terms inside the parentheses.
[tex]\[
(x - 3) - (x - 8) = x - 3 - x + 8
\][/tex]
2. Simplify both sides:
- LHS:
[tex]\[
4x + 4x - 10 = 8x - 10
\][/tex]
- RHS:
[tex]\[
x - 3 - x + 8 = 5
\][/tex]
3. Equate the simplified expressions:
[tex]\[
8x - 10 = 5
\][/tex]
4. Solve for x:
- Add 10 to both sides of the equation.
[tex]\[
8x - 10 + 10 = 5 + 10 \implies 8x = 15
\][/tex]
- Divide both sides by 8 to isolate x.
[tex]\[
x = \frac{15}{8} = 1.875
\][/tex]
So, the solution for x is [tex]\( x = 1.875 \)[/tex].
### Expression 2: Expanding [tex]\((2m - 3n)^2\)[/tex]
Given the expression:
[tex]\[ (2m - 3n)^2 \][/tex]
1. Use the distributive property (a - b)^2 = a^2 - 2ab + b^2:
- Here, [tex]\(a = 2m\)[/tex] and [tex]\(b = 3n\)[/tex].
2. Expand the expression:
[tex]\[
(2m - 3n)(2m - 3n)
\][/tex]
3. Distribute the terms:
[tex]\[
= (2m)(2m) - (2m)(3n) - (3n)(2m) + (3n)(3n)
\][/tex]
[tex]\[
= 4m^2 - 6mn - 6mn + 9n^2
\][/tex]
4. Combine like terms:
[tex]\[
= 4m^2 - 12mn + 9n^2
\][/tex]
So, the expanded form of [tex]\((2m - 3n)^2\)[/tex] is [tex]\( 4m^2 - 12mn + 9n^2 \)[/tex].
### Summary
- The solution for [tex]\( x \)[/tex] in the given equation is [tex]\( x = 1.875 \)[/tex].
- The expanded form of [tex]\( (2m - 3n)^2 \)[/tex] is [tex]\( 4m^2 - 12mn + 9n^2 \)[/tex].
