FACTORISING EQUATIONS

Transcription

1 STRIVE FOR EXCELLENCE TUTORING Factorising expressions with 2 terms FACTORISING EQUATIONS There are only 2 ways of factorising a quadratic with two terms: 1. Look for something in common between the 2 terms. Often this will be an x or a bracket (x+a) Eg: Factorise x 2 + 3x here, both terms have x =x(x+3) Eg: Factorise 3x(x-2) + 2(x-2) here, the bracket (x-2) is common =(x-2)(3x+2) 2. The other case is where nothing is common. Usually, one term will be an x 2 and the other term will be a number. This will be in the form a 2 -b 2. Recall that the difference of perfect squares a 2 -b 2 can be factorised into (a+b)(a-b) Example 1: Factorise x 2 9 You have to recognise that 9=3 2, and of course x 2 is a perfect square. We now use the difference of perfect square formula to factorise into (a+b)(a-b) x 2-9 = (x + 9 ) (x - 9 ) = (x+3) (x-3) Example 2: Factorise x 2 17 Even though 17 isn t a perfect square, we can still factorise it using the difference of perfect squares formula. We just don t get a nice number in the brackets. x 2 17 = (x + 17 ) (x - 17 ) Example 3: Factorise 4px 2 256p This is a combination of the two methods. Here, 4 is a common factor and p is common too, so the first step is to take out 4p. 4p(x 2 64) Now, recognise that inside the brackets we have a perfect square in the form a2 b2. We now use the difference of perfect squares formula to factorise this further: 4p(x + 8) (x 8)

2 Factorising expressions with 3 terms These are expressions of the form ax 2 + bx + c. When the co-efficient of x 2 is 1, this is straight forward. Example 1: Factorise x 2 + 3x + 2 Step 1: Find factors of the number (in this case the number is 2) The factors of 2 are 1 and 2. Step 2: Using the factors, try and add or subtract to make the co-efficient of x, which is 3 in this case. 2+1=3 Step 3: Open two pairs of brackets. ( ) ( ) Step 4: Write x as the first term in each bracket. (x ) (x ) Step 5: We wanted +2 and +1 to make 3, so we write them in the brackets in any order. (x + 2) (x + 1) x2 + 3x + 2 = (x + 2) (x + 1) Example 2: Factorise a 2-6a-7 Step 1: Factors of 7 are 1 and 7 Step 2: To make -6, we need Step 3: Open two brackets ( ) ( ) Step 4: Write x as the first term (x ) (x ) Step 5: Write in the -7 and the +1 (x-7) (x+1) Example 3: Factorise t 2-6t+8 Step 1: Factors of 8 are 4 and 2, 8 and 1 Step2: To make -6, we need -4 and -2 (we can t do anything with the 8 and 1 to make -6) Step 3: ( ) ( ) Step 4: (x ) (x ) Step 5: (x 4) (x 2) Continued next page

3 It becomes trickier when the co-efficient of x 2 is not 1. Example 1: Factorise 2x 2 +5x+2 Step1: Here, we need to find factors of both the number 2 and the co-efficient of x 2 (also 2 in this case). Step 2: Now, take one of the factors on the left and multiply it by one on the right. You do the same with the other pair. 2x 2 + 5x x 2 = 4 1 x 1 = 1 We get 4 and 1 which we can add to make 5 (the co-efficient of x) Step 3: Open the 2 brackets ( ) ( ) Step 4: Instead of writing x as the first term, we have to take into account the factors we used. We split the 2 in front of the x2 into 2x1 so we write 2x and 1x (2x ) (x ) Step 5: We paired the 2 with the other 2, and the 1 with the other 1. They go in opposite brackets. (2x 1) (x 2) Step 6: The sign is whatever sign the factors of the number (2) were. In this case they were both positive. (2x + 1) (x + 2)

4 Example 2: Factorise 15x 2 + x 2 Step 1: Factors: 15x 2 + x - 2 Step 2: Trial and error 15 x 2=30 1 x 1=1 can t make 1 15 x 1 =15 1 x 2=2 can t make 1 5 x 2=10 3 x 1=3 can t make 1 5 x 1=5 3 x 2=6 6 5=1 So, we want the 5 and 3 to go with the 2 and 1. The 5 goes with the 1 and the 3 goes with the 2. (5x 2) (3x 1) Step 3: We want to get positive 6 and negative 5. The 3x2 gave us 6, so both will be positive. The 5x1 gave us 5, so we need the 1 to be -1, so that we get -5 Step 4: Fill in the signs. Completing the square (5x + 2) (3x 1) In simple algebra, completing the square means to convert a quadratic equation. Example 1: Complete the square for x x Step 1: Take the co-efficient of x (10). Then divide it by 2 (5) and square it (25) Step 2: Add and subtract this number. X2 + 10x Step 3: The first 3 terms will factorise to give a perfect square. X (x + 5) (x + 5) -25 (x + 5)

5 Example 2: Complete the square for x 2-4x-7 Step 1: Co-efficient of x is -4. Divide by 2 = -2. Square this figure and that will equal = 4. Step 2: Add and subtract 4 x 2 4x (you write the +4 then the -4) Step 3: Factorise first three terms into a perfect square. x 2 4x = (x 2) (x 2) -11 = (x 2) 2-11