Following on from my 'solving quadratic equations' post from last week, i would like to introduce to all the Steemit learners out there, a much harder form. This is where we solve quadratic expressions in the form ax² +bx + c . These sorts of questions are different to the ones in the other blog in that the coefficient of the first term is greater than just x².
The method - step by step.
Example question: 2x² + 15x + 7
step 1: Multiply the first coefficient (in this case, 2) by the number at the end of the expression which is 7.
7x2=14
step 2: After multiplying the two numbers together. Find factors of the product and make sure they add to make the number in the centre of the expression. Here, the number at the centre of the expression is 15. So in this case, it would be 1 and 14 because 1x14=14 and 1+14=15.
step 3: copy out the expression BUT instead of it containing the '15x' like before, insert the new factors that we have but with an 'x' in front of each one.
2x² + 14x + 1x + 7
step 4: Factorise the first two terms of the expression.
2x² + 14x when factorised is... 2x (x+7)
step 5: replicate the bracket that you have gotten from factorising your first two terms.
equation: 2x² + 14x + 1x + 7
2x (x+7) ? (x+7)
step 6: what would '?' have to be for it to be equal to 1x+7. It would have to be 1
step 7: now you have got to 2x (x+7) +1(x+7) join up the terms outside the bracket to form its own separate bracket. Then have the (x+7) bracket as a separate bracket.
ANSWER: (2x +1) (x+7)
another practise question involving negatives
example question: 6x² + 5x - 6
step 1: As the method shows above. we take the coefficient of first term and multiply it by the last number.
6 x -6 = -36
step 2: Now we find the factors of -36 which also add to make +5.
the factors you will need are +9 and -4. This is because 9 x -4 = -36 and 9-4=5
step 3: Copy out the expression but instead of including the '+5x', insert the new factors that we have, 9 and -4, but with an 'x' in front of each one.
6x² + 9x -4x -6
step 4: As shown in the method, factorise the first two terms of the expression.
6x² +9x when factorised is... 3x(2x+3)
step 5: As shown in the method, replicate the bracket that you have gotten from factorising the first terms.
equation: 6x² + 9x -4x - 6
3x(2x+3) ?(2x+3)
step 6: What would '?' have to be for it to be equal to -4x - 6 . It would have to be -2.
step 7: Now that you have got to 3x(2x+3) -2 (2x+3) combine the two terms outside of the bracket to create its own separate bracket. Then have (2x+3) as a separate bracket.
Answer: (3x-2) (2x+3)
Practise questions for you to try... (the answers will be below)
1. 2x²-5x-12
2. 4x²+8x+3
3. 9x²-6x-8
4. 4x²-9x+5
5. 2x²+7x-4
6. 5x²+6x-8
answers
1. (2x-3) (x-4)
2. (2x+1) (2x+3)
3.(3x-4) (3x+2)
4.(4x-5) (x-1)
5. (2x-1) (x-4)
6. (5x-4) (x+2)
Hope this was useful to you.
Hope everyone is having a great weekend.
Jonathan.