Quadratic Equations part A

Quadratic equations can be puzzling at first, but it is so easy. The only really ‘difficulty’, if you want to believe that there is actually a difficulty, is finding the split numbers. Split numbers is a term I use, so don’t be scratching your head thinking what split numbers are. Well first you have to identify what a quadratic equation is. In “Linear Equation”, the topic below this one, I demonstrated how to identify a quadratic equation. Now let’s look at the typical quadratic equation:

1) 3x^2 - 7x - 6 = 0
To begin any quadratic equation you must have 4 terms in the equation. All quadratic equations you see come with 3 terms like the question above. You have to create another term by doing my split number method. The split number only involves altering the x term (e.g. -7x, as in the question above.)

ax^2 + bx +c = 0

1. the first step is to multiply the coefficient of x^2, a, (in this question, the coefficient is 3) by the constant, c, (in this question, the constant is -6). Ok I am going to put it in letter form so you will not get confuse.
the first step result is ………..a x c ₌ ac

2. The second step is to number split. You must get the correct numbers so that it satisfies this condition…..a + c = b. so let’s take a look of all the factors of (a x c)
a x c = 3 x (-6) = -18……now let’s look at the factors of -18

product …condition... (a x c) sum…..condition... (b)

1. (+18) x (-1) = - 18..............................(+18) + (-1) = 17
2. (-18) x (+1) = -18……………….............(-18) + (+1) = -17
3. (+9) x (-2) = -18…………………...........(+9) + (-2) = +7
4. (-9) x (+2) = -18…………………...........(-9) + (+2) = -7
5. (+6) x (-3) = -18…………………...........(+6) + (-3) = +3

6. (-6) x (+3) = -18…………………...........(-6) + (+3) = -3

These are all the factors of -18. Now we must look for the line that satisfies the condition …. a x c = b. there is only one line where this condition meets, in any quadratic equation. There is only one rare case know of, which involves the number 6 and in my next session I will talk about it in more details. In this case the only line which satisfies the condition and its line 4. In line 4, the product is -18 and the sum is -7, exactly what we need.

3) *Using the sum condition in line 4, replace -7x with (-9 +2)x, solve for x ( you must get two values). Now we have completed the number splitting, now rewrite the equation. NOTE: the equation has not been altered in terms of its numeric value, it is the exact equation. The equation before was

3x^2 -7x -6 =0………now it is…..3x^2 + (-9 + 2)x - 6 = 0
…………………………………............3x^2 – 9x +2x – 6 = 0

Now factorize…………………........3x(x – 3) + 2(x – 3) = 0

Now group…………………….........(3x + 2)(x – 3) = 0
Now solve individually…………...(3x + 2) = 0 (x – 3) = 0
(solve it like a linear equation)……………x = -2/3……x = 3

You now know how to solve quadratic equations. Now you need practice. For those who don’t know how to factorize look for future topics on my site, so you can learn, then you can fully understand this topic.

*DO NOT ADD BACK (-9 +2) TO GET BACK (-7) AS YOU WILL DEFEAT THE PURPOSE OF NUMBER SPLITTING, YOU SPLIT (-7) TO GET (-9 +2) SO THAT YOU HAVE FOUR TERMS IN THE QUADRATIC EQUATION.