How do you factor numbers?
2x squared + 5x-3 for instance?
2x^2 + 5x -3
Make two sets of parentheses.
( ) ( )
The first number in each set must be such that the product will be 2x^2. Obviously, that must be 2xand 1x so put them in.
(2x )(x )
The next thing in the set is the sign. That means the sign of one must be - and the sign of the other must be + because a + number times a - number is the only way to get a - number; i.e., two - numbers give + and two + numbers give a +. We may need to change the order later but this is as good a place to start as any. So put those in.
(2x + ?)(x - ?)
That leaves us with the question marks.
We want a number that adds to 3 but gives 5 when multiplied by 2 and 1. For the number 3 there are only 3*1 but it makes a difference where the 3 and 1 are placed. We just try it and see. Try it as 3 in the first and 1 in the second.
(2x+3)(x-1)
Now let's check it.
The first number is 2x*x = 2x^2. check.
The last number is +3*-1=-3. check.
The middle number is made up of the sum of the products of the "inner two" and the "outer two."
The outer first: 2x*-1=-2x
The inner next: 3*x=3x
Then -2x+3x=x so this can't be right. We want it to be +5x.
Then try reversing the signs and see if that works. It won't.
Then try reversing the signs AND the numbers and try that.
I will leave it to you at that but post back if you need further assistance. Be sure to tell us what you don't understand. Practice makes perfect. If you do enough of these you get to the point that you can figures most of them in your head and get it right the first or second time. I made it wrong the first two times on purpose so you could see how it was done.
2xsquared +5x-3
2x^2-5x+1
9x^2+12x+4
To factor a quadratic expression like 2x^2 + 5x - 3, we can follow these steps:
1. Write down the expression: 2x^2 + 5x - 3.
2. Make two sets of parentheses: ( ) ( ).
3. The first number in each set should be selected so that their product is equal to 2x^2. In this case, the factors of 2x^2 are (2x)(x), so we place them in the parentheses: (2x )(x ).
4. The next step is to choose signs for the two numbers in each set. We need a combination that will give us +5x when multiplied. Since we have a positive and a negative sign in the expression, we can try placing a positive sign in front of one number and a negative sign in front of the other: (2x + ?)(x - ?).
5. We are left with the question marks. We need to find two numbers that multiply to give -3 and add up to 5. The possible combinations are 3 and -1 or -3 and 1. We try the first combination, placing 3 in the first set and -1 in the second set: (2x + 3)(x - 1).
6. Now let's check if the factors are correct. Multiply the first numbers in each set: 2x * x = 2x^2 (check). Multiply the last numbers in each set: 3 * -1 = -3 (check).
7. To check the middle term, we multiply the two "inner" numbers and the two "outer" numbers, then combine them. The outer product is 2x * -1 = -2x, and the inner product is 3 * x = 3x. The sum of the outer and inner products is -2x + 3x = x. Since this matches the original expression's middle term of +5x, we have factored correctly.
Therefore, the factored form of 2x^2 + 5x - 3 is (2x + 3)(x - 1).