{"id":223,"date":"2023-01-30T17:10:31","date_gmt":"2023-01-30T17:10:31","guid":{"rendered":"https:\/\/cwassignments.com\/blog\/?p=223"},"modified":"2024-05-10T07:50:29","modified_gmt":"2024-05-10T07:50:29","slug":"solving-by-factoring","status":"publish","type":"post","link":"https:\/\/cwassignments.com\/blog\/solving-by-factoring\/","title":{"rendered":"Solving by Factoring"},"content":{"rendered":"

Solving a quadratic (or any kind of equation) by factoring it makes use of a principle known as the zero-product
\nrule.<\/p>\n\n\n\n
\n

Zero Product Rule<\/h4>\n

If\u00a0ab<\/em>\u00a0=\u00a00 then either\u00a0a<\/em>\u00a0=\u00a00 or\u00a0b<\/em>\u00a0= 0 (or both).<\/p>\n

In other words, if the product of two things is zero then one of those two things must be zero, because the only way to multiply something and get zero is to multiply it by zero.<\/p>\n

Thus, if you can factor an expression\u00a0that is equal to zero<\/u>, then you can set each factor equal to zero and solve it for the unknown.<\/p>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0The expression\u00a0must\u00a0<\/em>be set equal to zero to use this principle<\/p>\n

\u00b7\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0You can always make any equation equal to zero by moving all the terms to one side.<\/p>\n

Example:<\/strong><\/b><\/p>\n\n\n\n\n\n\n\n
Given:<\/td>\nx<\/em>2<\/sup>\u00a0\u2013\u00a0x<\/em>\u00a0= 6<\/td>\n<\/tr>\n
Move all terms to one side:<\/td>\nx<\/em>2<\/sup>\u00a0\u2013\u00a0x<\/em>\u00a0\u2013 6 = 0<\/td>\n<\/tr>\n
Factor:<\/td>\n(x<\/em>\u00a0\u2013 3)(x<\/em>\u00a0+ 2) = 0<\/td>\n<\/tr>\n
Set each factor equal to zero and solve:<\/td>\n(x<\/em>\u00a0\u2013 3)\u00a0= 0\u00a0\u00a0\u00a0OR<\/strong><\/b>\u00a0\u00a0\u00a0(x<\/em>\u00a0+ 2) = 0<\/td>\n<\/tr>\n
Solutions:<\/td>\nx\u00a0<\/em>= 3\u00a0OR<\/strong><\/b>\u00a0\u00a0\u00a0x<\/em>\u00a0=\u00a0\u20132<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

No Constant Term<\/h3>\n

If a quadratic equation has no constant term (i.e.\u00a0c<\/em>\u00a0= 0) then it can easily be solved by factoring out the common\u00a0x<\/em>\u00a0from the remaining two terms:<\/p>\n

Then, using the zero-product rule, you set each factor equal to zero and solve to get the two solutions:<\/p>\n

x<\/em>\u00a0=\u00a00\u00a0\u00a0\u00a0or<\/strong><\/b>
\nax<\/em>\u00a0+\u00a0b<\/em>\u00a0=\u00a00<\/p>\n\n\n\n
x\u00a0=\u00a0<\/em>0\u00a0\u00a0\u00a0or<\/strong><\/b>\u00a0\u00a0\u00a0x<\/em>\u00a0=\u00a0\u2013b<\/em>\/a<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

WARNING<\/strong><\/b>:
\nDo not divide out the common factor of\u00a0x<\/em>\u00a0or you will lose the\u00a0x<\/em>\u00a0= 0 solution. Keep all the factors and use the zero-product rule to get the solutions.<\/p>\n

Trinomials<\/h3>\n

When a quadratic has all three terms, you can still solve it with the zero-product rule if you are able to factor the trinomial.<\/p>\n