Title: 7. Removing brackets
Course: Demonstrating STACK (v4.3.7)

Expand the following:

\(\left( x-[value] \right) \left(x+[value] \right)\) =
Feedback: The difference of two squares comes up very often in mathematics: \[\left( x-y \right) \left(x+y \right) = x^2-y^2. \] In this example: \[ \left( x-[value] \right) \left( x+[value] \right) = x^2-[value]. \]

Expand the following:

\(\left( [value] \right)^2\) = .
Feedback: The following formula for squaring \((a+b)\) is very important in mathematics: \[\left( a+b \right)^2 = a^2+2ab+b^2.\] Hence, in this example: \[[value] = [value].\]

Expand the following:

\(\left( [value] \right) \left( [value] \right)\) = .
Feedback: There are a number of ways to multiply out brackets. One way is to follow the general equation: \[ \left(a_1+a_2 \right) \left(b_1+b_2 \right) = a_1\left(b_1+b_2 \right)+a_2\left(b_1+b_2 \right) = a_1b_1+a_1b_2+a_2b_1+a_2b_2. \] In this example: \[\left( [value] \right) \left([value] \right) = [value]+\left([value] \right)+\left( [value] \right)+ \left([value] \right) = [value].\]

Expand the following bracket: \[[value].\]
Feedback: In this example multiply out the bracket using the following rule: \[ a\left(b+c \right) = ab + ac. \] Be careful that you get the correct sign when negative numbers are involved (remembering the rules for multiplying negative numbers). In this example: \[[value] = \left([value]\right)\cdot \left([value]\right)+\left([value] \right)\cdot \left([value] \right) = [value].\]

Expand the following bracket: \[[value].\]
Feedback: When you have more than two terms inside a bracket or more than one variable, the process of expanding out brackets is no different. You simply need to remember to multiply everything in the one bracket, by everything in the other. In this example: \[[value] = xy\cdot \left([value] \right)+xy\cdot \left([value] \right)+ xy \cdot \left([value] \right) = [value].\]

Expand the following:

\(\left( [value] \right) \left([value] \right)\) = .
Feedback: There are a number of ways to multiply out brackets. One way is to follow the general equation: \[ \left(a_1+a_2 \right) \left(b_1+b_2 \right) = a_1\left(b_1+b_2 \right)+a_2\left(b_1+b_2 \right) = a_1b_1+a_1b_2+a_2b_1+a_2b_2. \] In this example: \[ \begin{align} \left( [value] \right) \left( [value] \right) &= [value]+\left( [value] \right)+ \left([value] \right)+\left( [value] \right), \\ &= [value]. \end{align} \]

Expand the following:

\( \left( [value] \right) \left( [value] \right)\) =
Feedback: There are a number of ways to multiply out brackets. One way is to follow the general equation: \[ \left(a_1+a_2 \right) \left(b_1+b_2 \right) = a_1\left(b_1+b_2 \right)+a_2\left(b_1+b_2 \right) = a_1b_1+a_1b_2+a_2b_1+a_2b_2. \] In this example: \[\begin{align} [value] &= [value]+\left([value]\right)+\left([value]\right)+\left([value]\right) \\ &= [value]. \end{align} \]

Expand the following:

\(2x \left( [value] \right) \left( [value] \right)\) = .
Feedback: When multiplying out 3 brackets the best thing to do is focus on 2 brackets at a time. In this example we multiply out the \(\left( [value] \right) \left( [value] \right)\) factor first putting the \(2x\) to the side for the time being: \[[value].\] Now we just multiply the remaining two brackets as normal: \[[value].\]

Expand the following:

\(\left([value]\right)\left([value]\right)\left([value]\right)\) =
Feedback: When multiplying out 3 brackets the best thing to do is focus on 2 brackets at a time. In this example we multiply out the \(\left([value]\right)\left([value]\right)\) factors first putting the \( \left( [value] \right) \) to the side for the time being: \[[value].\] Now we just multiply the remaining two brackets as normal: \[[value].\]

Expand the following: \[ [value]\]
Feedback: When multipling out brackets involving surds the method is very much the same. Just remember that \[ \sqrt{x}\sqrt{x}=x^{\frac{1}{2}}\times x^{\frac{1}{2}} = x^{\frac{1}{2}+\frac{1}{2}}=x.\] In this example: \[[value] = [value]-[value]\sqrt[value]+[value]\sqrt[value]-[value] = [value]. \]