Solving quadratic equations by factoring
In order to solve a quadratic equation by factoring, follow these steps:
1. Put the equation in the standard form (ax2+bx+c=0)
2. Factor the equation (find two numbers that will not only multiply to equal the constant term "c", but also add up to equal "b", the coefficient on the x-term).
3. Set each of the two binomial expressions equal to cero.
4.
Solve each of the equations.
5. Check you answer.
Solve by factoring x2+2x = 15
1. Put the equation in the standard form: x
2+2x-15=0
2.
We require to numbers that multiply together to give -15 and add together to give -2.
x2+2x-15=(x-3)(x+5).
3. Set each of the two binomial expressions equal to cero.
x2+2x-15 = 0
(x-3)(x+5) =0
4. Solve each equation:
x-3=0, x=3
x+5=0, x=-5
When
the leading coefficient (the number on the x2 term) is not 1, the first step in factoring will be to multiply "a" and "c"; then we'll need to find factors of the product "ac" that add up to "b".
Solve by factoring 2x2+4x-6 = 0
We need to find factors of 12 (ac=2·(-6)=-12) that add up to +4.
We will use the pair "-2 and 6".
Draw a two-by-two grid, putting the first term in the upper left-hand corner and the last term in the lower right-hand corner:
Take the factors –2 and 6 and put them, complete with their signs and variables, in the diagonal corners:
Factor the rows and columns:
Then, 2x
2+4x-6 = (2x-2)(2x+6)
We will find the solutions of the equations by solving each equation:
2x-2=0, x=1
2x+6=0, x=-3
Sometimes you can not find integer factors that work, then this quadratic is said to be "
unfactorable over the integers" or "
prime". On these cases, you must try to solve the equation using another method (the quadratic formula,...)
