How do I know which factors to use when solving a quadratic equation?

Question

Example: 2x^2-7x-30

I tried to use -10 and +3, which also adds to -7 (the middle term). This was incorrect and I found out the correct numbers were -5 and 6. I know that these multiply to 30, but how do I know that these two numbers are the factors are the ones that solve the quadratic equation? Do I always have to plug in the numbers into the equation to check if it solves for 0? That would take a long time for numbers that have lots of multiplication combinations.

Answer 1

Yes, it takes a long time, but como is correct - there are four different factorisations of 30, and each one has to be tried each way round with the one and only factorisation (2x * x) of 2x², so it is trial and error, with eight trials and seven errors.

Answer 2

The reason your first method didnt work is because the coeffecient of the x squared is 2, not 1.

Solving quadratics can be tricky, and you should always check your answers, but there is a simple to use equation that will give you the right answer [even when the solutions are imaginary or irrational], its called the quadratic formula:

http://mathworld.wolfram.com/QuadraticFormula.html

very useful.

Answer 3

The only possible factors of 2x² are 2x and x so this is fixed starting point:-
(2x -------).(x--------)
Now look at factors of 30:-
6,5---5,6
10,3---3,10
15,2---2,15
30,1--1,30
Now it is trial and error with signs to find -7x:-
This gives 5 , - 6
This gives 5x - 12x = - 7x i.e.:-
(2x + 5).(x - 6)
Check
2x.(x - 6) + 5.(x - 6)
= 2x² - 12x + 5x - 30
= 2x² - 7x - 30
P.S. you do not have an equation.
An equation would be:-
2x² - 7x - 30 = 0 and then
(2x + 5).(x - 6) = 0
x = - 5/2, x = 6

Answer 4

Use the quadratic formula.

from ax^2 + bx + c

X = (-b +/- ((b^2 -4ac)^(1/2)))/2a

In your equation above
a = 2
b = -7
c = -30

so X will be -2.5 and 6.

And yes to check if it solves the equation you have to substitute each number to the equation. That is if you are unsure of your computation only.