Algebraic techniques and factorising · Years 9–10
In Algebraic techniques and factorising (Years 9–10) you practise 6 skills: Expanding double brackets · Special products: squares and the difference of two squares · Equations with brackets to expand · Comparing areas · Factorising: taking out a common factor · Factorising quadratics. To expand, multiply each term in the first bracket by each term in the second, then collect like terms. The special products save the long multiplication: the square of a sum is the first term squared, plus twice the product of the two terms, plus the second term squared. In the square of a difference the middle term is subtracted, and a sum times a difference of the same two terms is the first squared minus the second squared.
Factorising is the reverse of expanding: write an expression as a product. Always start by taking out the largest common factor. To factorise a quadratic whose squared term has coefficient 1, look for two numbers that multiply to the number term and add to the coefficient of the middle term, and check by expanding back. In an equation with products of brackets, expand the brackets first; when the squared terms cancel, a linear equation is left. A quadratic equation that factorises is solved another way: a product is 0 only when one of its factors is 0, so set each factor to 0 and find one or two solutions. In an algebraic fraction, factorise the numerator and the denominator and cancel a common factor. To multiply fractions, multiply numerator by numerator and denominator by denominator; to divide, multiply by the reciprocal. A value that makes a denominator 0 is excluded. In context: a carpenter compares a rectangular board with a square one, and expanding the brackets shows which is larger, and by how much, without knowing the length itself.
Tip: Check your expansion: put a small number, such as 1, into both the original expression and your answer. If they give the same number, you are probably right.
What you learn
Expanding double brackets
For example: Expand and simplify: (x + 2)(x + 9)
Answer and solution
The answer: x² + 11x + 18
The four products: x² + 9x + 2x + 18 Collect like terms: x² + 11x + 18
Special products: squares and the difference of two squares
For example: Expand and simplify: (x − 7)²
Answer and solution
The answer: x² − 14x + 49
The square of a difference: the first term squared, minus twice the product of the two terms, plus the second term squared: x² − 2 × x × 7 + 7² So: x² − 14x + 49
Equations with brackets to expand
For example: Solve: (x − 7)(x + 9) = x² − 93
Answer and solution
The answer: −15
Expand the brackets: x² + 2x − 63 = x² − 93 Subtract x² from both sides and rearrange: 2x = −30 Divide by 2: x = −15
Comparing areas
For example: A rectangle has sides x + 3 and x + 7 , and a square has side x + 5 . How much larger is the area of the square than the area of the rectangle?
Answer and solution
The answer: 4
The area of the rectangle: (x + 3)(x + 7) = x² + 10x + 21 The area of the square: (x + 5)² = x² + 10x + 25 The difference: (x² + 10x + 25) − (x² + 10x + 21) = 4
Factorising: taking out a common factor
For example: Factorise fully: 10x + 4
Answer and solution
The answer: 2(5x + 2)
The common factor: 2 What is left in the bracket: 5x + 2 Check by expanding back to the original expression: 2(5x + 2) = 10x + 4
Factorising quadratics
For example: Factorise: x² + 11x + 18
Answer and solution
The answer: (x + 2)(x + 9)
The two numbers: 2 × 9 = 18 2 + 9 = 11 So: x² + 11x + 18 = (x + 2)(x + 9)
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the national curriculum in England: mathematics (key stages 3 and 4).