Sure, let's solve these problems step by step.
### Question 6:
To find the value of [tex]\( p \)[/tex] such that the expression [tex]\( 3x^2 + 4x - p \)[/tex] equals 5, we need to solve the equation
[tex]\[ 3x^2 + 4x - p = 5 \][/tex]
for [tex]\( p \)[/tex] given specific values of [tex]\( x \)[/tex].
#### Part (a): When [tex]\( x = -1 \)[/tex]
1. Substitute [tex]\( x = -1 \)[/tex] into the equation:
[tex]\[
3(-1)^2 + 4(-1) - p = 5
\][/tex]
2. Simplify the left side:
[tex]\[
3(1) + 4(-1) - p = 5
\][/tex]
[tex]\[
3 - 4 - p = 5
\][/tex]
3. Combine like terms:
[tex]\[
-1 - p = 5
\][/tex]
4. Isolate [tex]\( p \)[/tex] by adding 1 to both sides:
[tex]\[
-p = 6
\][/tex]
5. Multiply both sides by -1:
[tex]\[
p = -6
\][/tex]
So, when [tex]\( x = -1 \)[/tex], [tex]\( p = -6 \)[/tex].
#### Part (b): When [tex]\( x = 1 \)[/tex]
1. Substitute [tex]\( x = 1 \)[/tex] into the equation:
[tex]\[
3(1)^2 + 4(1) - p = 5
\][/tex]
2. Simplify the left side:
[tex]\[
3(1) + 4(1) - p = 5
\][/tex]
[tex]\[
3 + 4 - p = 5
\][/tex]
3. Combine like terms:
[tex]\[
7 - p = 5
\][/tex]
4. Isolate [tex]\( p \)[/tex] by subtracting 7 from both sides:
[tex]\[
-p = -2
\][/tex]
5. Multiply both sides by -1:
[tex]\[
p = 2
\][/tex]
So, when [tex]\( x = 1 \)[/tex], [tex]\( p = 2 \)[/tex].
### Question 7:
We need to find the value of the expression [tex]\( a^3 - b^3 + 3ab(a - b) \)[/tex] when [tex]\( a = -3 \)[/tex] and [tex]\( b = 1 \)[/tex].
1. Substitute [tex]\( a = -3 \)[/tex] and [tex]\( b = 1 \)[/tex] into the expression:
[tex]\[
(-3)^3 - (1)^3 + 3(-3)(1)((-3) - 1)
\][/tex]
2. Calculate each term:
[tex]\[
(-3)^3 = -27
\][/tex]
[tex]\[
(1)^3 = 1
\][/tex]
3. Calculate [tex]\( 3ab(a - b) \)[/tex]:
[tex]\[
3(-3)(1)((-3) - 1)
\][/tex]
4. Calculate inside the parentheses:
[tex]\[
(-3) - 1 = -4
\][/tex]
5. Multiply the terms:
[tex]\[
3(-3)(1)(-4) = 3(-3)(-4) = 3 \times 12 = 36
\][/tex]
6. Combine all terms:
[tex]\[
-27 - 1 + 36
\][/tex]
7. Simplify:
[tex]\[
-28 + 36 = 8
\][/tex]
So, the value of the expression when [tex]\( a = -3 \)[/tex] and [tex]\( b = 1 \)[/tex] is 8.
### Summary:
- For [tex]\( x = -1 \)[/tex], [tex]\( p = -6 \)[/tex].
- For [tex]\( x = 1 \)[/tex], [tex]\( p = 2 \)[/tex].
- The value of the expression [tex]\( a^3 - b^3 + 3ab(a - b) \)[/tex] with [tex]\( a = -3 \)[/tex] and [tex]\( b = 1 \)[/tex] is 8.
