4A - Expanding quadratics
Expanding brackets involves removing the brackets by multiplying each term in the brackets by every term outside of the brackets. The process of expansion is also known as the distributive law. Once an expression has been expanded, it may be possible to simplify by collecting like terms. Recall that like terms contain the same variables raised to the same power. For example, \(7x\) and \(-2x\) are like terms.
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■ Warning! When expanding brackets be very careful if any of the terms contain a negative sign. |
Expanding with only one term outside the brackets:
When there is only one term outside the brackets the process of expansion is relatively simple. In this case we multiply each term inside the bracket by the single term outside of the bracket. For example: \[k(a+b)=ka+kb\]
When there is only one term outside the brackets the process of expansion is relatively simple. In this case we multiply each term inside the bracket by the single term outside of the bracket. For example: \[k(a+b)=ka+kb\]
4A - Example 1: Expanding with only one term outside the brackets
Expand the following expression: \[4(3x-1)\]
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4A - Example 1: Video solution
4A - Example 1: Practice
Question 1: Expand \(5(3+2x)\) Question 2: Expand \(-4(7x+y)\) 4A - Example 1: Solutions
Question 1: \(5(3+2x)=15+10x\) Question 2: \(-4(7x+y)=-28x-4y\) |
4A - Example 2: Expanding with only one term outside the brackets
Expand the following expression: \[-2x(x-7)\]
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4A - Example 2: Video solution
4A - Example 2: Practice
Question 1: Expand \(x(3-x)\) Question 2: Expand \(-3x(4+5x)\) 4A - Example 2: Solutions
Question 1: \(x(3-x)=3x-x^2\) Question 2: \(-3x(4+5x)=-12x-15x^2\) |
Expanding when there are two brackets present:
Expanding becomes more difficult when more terms are outside of the bracket but still need to be multiplied through. When this is the case you can use the box/grid method or FOIL to ensure you expand the set of brackets properly. In general: \[(a+b)(c+d)=ac+ad+bc+bd\]
Expanding becomes more difficult when more terms are outside of the bracket but still need to be multiplied through. When this is the case you can use the box/grid method or FOIL to ensure you expand the set of brackets properly. In general: \[(a+b)(c+d)=ac+ad+bc+bd\]
4A - Example 3: Using grids/boxes to expand brackets
Expand the following expression using the box method: \[(x+4)(x-3)\]
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4A - Example 3: Video solution
4A - Example 3: Practice
Question 1: Expand \((x+7)(x+2)\) Question 2: Expand \((3-x)(x+2)\) 4A - Example 3: Solutions
Question 1: \((x+7)(x+2)=x^2+9x+14\) Question 2: \((3-x)(x+2)=-x^2+x+6\) |
4A - Example 4: Using grids/boxes to expand brackets
Expand the following expression using the box method: \[(6x+2)(4x-3)\]
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4A - Example 4: Video solution
4A - Example 4: Practice
Question 1: Expand \((2x+1)(x+6)\) Question 2: Expand \((5x+3)(2x+5)\) 4A - Example 4: Solutions
Question 1: \((2x+1)(x+6)=2x^2+13x+6\) Question 2: \((5x+3)(2x+5)=10x^2+31x+15\) |
4A - Example 5: Using FOIL to expand brackets
Expand the following expression using FOIL: \[(x+2)(x-5)\]
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4A - Example 5: Video solution
4A - Example 5: Practice
Question 1: Expand \((x+5)(x-3)\) Question 2: Expand \((-x+4)(x+9)\) 4A - Example 5: Solutions
Question 1: \((x+5)(x-3)=x^2+2x-15\) Question 2: \((-x+4)(x+9)=-x^2-5x+36\) |
4A - Example 6: Using FOIL to expand brackets
Expand the following expression using FOIL: \[(4x+3)(2x+4)\]
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4A - Example 6: Video solution
4A - Example 6: Practice
Question 1: Expand \((9x+4)(x-3)\) Question 2: Expand \((3x+8)(2x+5)\) 4A - Example 6: Solutions
Question 1: \((9x+4)(x-3)=9x^2-23x-12\) Question 2: \((3x+8)(2x+5)=6x^2+31x+40\) |
Patterns associated with expansion:
There are two important patterns associated with expansion: perfect squares and the difference of perfect squares.
Case 1: Expanding using perfect squares
There are two important patterns associated with expansion: perfect squares and the difference of perfect squares.
Case 1: Expanding using perfect squares
\[(a+b)^2\] \[=a^2+2ab+b^2\]
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\[(a-b)^2\] \[=a^2-2ab+b^2\]
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■ Warning! A common error is for students to forget the middle term, it is important to note that \((a+b)^2\) is not equal to \(a^2+b^2\). |
4A - Example 7: Expanding using perfect squares
Expand the following expression: \[(x+1)^2\]
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4A - Example 7: Video solution
4A - Example 7: Practice
Question 1: Expand \((x+8)^2\) Question 2: Expand \((x-3)^2\) 4A - Example 7: Solutions
Question 1: \((x+8)^2=x^2+16x+64\) Question 2: \((x-3)^2=x^2-6x+9\) |
4A - Example 8: Expanding using perfect squares
Expand the following expression: \[(-5x+12)^2\]
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4A - Example 8: Video solution
4A - Example 8: Practice
Question 1: Expand \((3x+4)^2\) Question 2: Expand \((3-7x)^2\) 4A - Example 8: Solutions
Question 1: \((3x+4)^2=9x^2+24x+16\) Question 2: \((3-7x)^2=9-42x+49x^2\) |
Case 2: Expanding using the difference of perfect squares (DOPS) \[(a+b)(a-b)=a^2+ab-ab-b^2=a^2-b^2\]
4A - Example 9: Expanding using the difference of perfect squares (DOPS)
Expand the following expression: \[(x+4)(x-4)\]
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4A - Example 9: Video solution
4A - Example 9: Practice
Question 1: Expand \((x-11)(x+11)\) Question 2: Expand \((3-x)(3+x)\) 4A - Example 9: Solutions
Question 1: Expand \((x-11)(x+11)=x^2-121\) Question 2: Expand \((3-x)(3+x)=9-x^2\) |
4A - Example 10: Expanding using the difference of perfect squares (DOPS)
Expand the following expression: \[(2x+5)(2x-5)\]
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4A - Example 10: Video solution
4A - Example 10: Practice
Question 1: Expand \((4+5x)(4-5x)\) Question 2: Expand \((3x+\sqrt{5})(3x-\sqrt{5})\) 4A - Example 10: Solutions
Question 1: \((4+5x)(4-5x)=16-25x^2\) Question 2: \((3x+\sqrt{5})(3x-\sqrt{5})=9x^2-5\) |
More complex expansions:
Case 1: Expanding brackets with more than two terms inside
When brackets contain more than two terms you must be careful to multiply each term inside the bracket by every term outside of the bracket. Generally, it is best to use a grid/box to conduct these expansions.
Case 1: Expanding brackets with more than two terms inside
When brackets contain more than two terms you must be careful to multiply each term inside the bracket by every term outside of the bracket. Generally, it is best to use a grid/box to conduct these expansions.
4A - Example 11: Expanding brackets with more than two terms inside
Expand the following expression: \[(2x-3)(x^2-3x+4)\]
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4A - Example 11: Video solution
4A - Example 11: Practice
Question 1: Expand \((x+2)(x^2+5x+6)\) Question 2: Expand \((1-3x)(x^2+9x+14)\) 4A - Example 11: Solutions
Question 1: \((x+2)(x^2+5x+6)=x^3+7x^2+16x+12\) Question 2: \((1-3x)(x^2+9x+14)=-3x^3-26x^2-33x+14\) |
Case 2: Expanding when more than two sets of brackets are present
When more than two sets of brackets are present:
When more than two sets of brackets are present:
- Expand two sets of brackets and then place the expanded form in another set of brackets.
- Multiply each of the terms in the expanded form by the terms in the remaining bracket.
- Repeat, if necessary, until all brackets have been expanded.
4A - Example 12: Expanding brackets with more than two terms inside
Expand the following expression: \[(x-1)(x+2)(x+3)\]
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4A - Example 12: Video solution
4A - Example 12: Practice
Question 1: Expand \((x-3)(x+4)(x+5)\) Question 2: Expand \(2(1-3x)(x+2)(x-4)\) 4A - Example 12: Solutions
Question 1: \((x-3)(x+4)(x+5)=x^3+6x^2-7x-60\) Question 2: \(2(1-3x)(x+2)(x-4)=-6x^3+14x^2+44x-16\) |