{"id":309,"date":"2023-01-30T18:51:04","date_gmt":"2023-01-30T18:51:04","guid":{"rendered":"https:\/\/cwassignments.com\/blog\/?p=309"},"modified":"2024-05-01T13:34:55","modified_gmt":"2024-05-01T13:34:55","slug":"quadratic-formula","status":"publish","type":"post","link":"https:\/\/cwassignments.com\/blog\/quadratic-formula\/","title":{"rendered":"The Quadratic Formula"},"content":{"rendered":"

he solutions to a quadratic equation can be found directly from the quadratic formula.<\/p>\n\n\n\n
The equation<\/p>\n

ax<\/em>2<\/sup>\u00a0+\u00a0bx<\/em>\u00a0+
\nc<\/em>\u00a0= 0<\/p>\n

has solutions<\/p>\n

\"\"<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

The advantage of using the formula is that it always works. The disadvantage is that it can be more time-consuming than some of the methods previously discussed. As a general rule you should look at a quadratic and see if it can be solved by taking square roots; if not, then if it can be easily factored; and finally use the quadratic formula if there is no easier way.<\/p>\n

    \n
  • \u00a0 \u00a0 \u00a0Notice the plus-or-minus symbol (\u00b1) in the formula. This is how you get the two different solutions\u2014one using the plus sign, and one with the minus.<\/li>\n
  • \u00a0 \u00a0 \u00a0Make sure the equation is written in standard form before reading off\u00a0a<\/em>,\u00a0b<\/em>, and\u00a0c<\/em>.<\/li>\n
  • \u00a0 \u00a0 \u00a0Most importantly, make sure the quadratic expression is equal to zero.<\/li>\n<\/ul>\n

    The Discriminant<\/h3>\n

    The formula requires you to take the square root of the expression\u00a0b<\/em>2\u00a0<\/sup>\u2013 4ac<\/em>, which is called the\u00a0discriminant<\/em>\u00a0because it determines the nature of the solutions. For example, you can\u2019t take the square root of a negative number, so if the discriminant is negative then there are no solutions.<\/p>\n\n\n\n\n\n
    If\u00a0b<\/em>2<\/sup>\u00a0\u2013\u00a04ac<\/em>\u00a0>\u00a00<\/td>\nThere are two distinct real roots<\/td>\n<\/tr>\n
    If\u00a0b<\/em>2<\/sup>\u00a0\u2013\u00a04ac<\/em>\u00a0=\u00a00<\/td>\nThere is one real root<\/td>\n<\/tr>\n
    If\u00a0b<\/em>2<\/sup>\u00a0\u2013\u00a04ac<\/em>\u00a0<\u00a00<\/td>\nThere are no real roots<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

    Deriving the Quadratic Formula<\/h3>\n

    The quadratic formula can be derived by using the technique of completing the square on the general quadratic formula:<\/p>\n\n\n\n\n\n\n\n\n\n\n
    Given: \"\"<\/td>\n<\/td>\n<\/tr>\n
    Divide through by\u00a0a<\/em>:\"\"<\/td>\n<\/td>\n<\/tr>\n
    Move the constant term to the right
    \nside:\"\"<\/td>\n    		<\/td>\n<\/tr>\n
    Add the square of one-half the
    \ncoefficient of\u00a0x<\/em>\u00a0to both sides:\"\"<\/td>\n    		<\/td>\n<\/tr>\n
    Factor the left side (which is now a
    \nperfect square), and rearrange the right side:\"\"<\/td>\n    		<\/td>\n<\/tr>\n
    Get the right side over a common
    \ndenominator:\"\"<\/td>\n    		<\/td>\n<\/tr>\n
    Take the square root of both sides
    \n(remembering to use plus-or-minus):\"\"<\/td>\n    		<\/td>\n<\/tr>\n
    Solve for\u00a0x<\/em>:\"\"<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"

    he solutions to a quadratic equation can be found directly from the quadratic formula. The equation ax2\u00a0+\u00a0bx\u00a0+ c\u00a0= 0 has solutions The advantage of using the formula is that it always works. The disadvantage is that it can be more time-consuming than some of the methods previously discussed. As a general rule you should look […]<\/p>\n","protected":false},"author":1,"featured_media":686,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"_swt_meta_header_display":false,"_swt_meta_footer_display":false,"_swt_meta_site_title_display":false,"_swt_meta_sticky_header":false,"_swt_meta_transparent_header":false,"footnotes":""},"categories":[7],"tags":[],"class_list":["post-309","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-algebra"],"_links":{"self":[{"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/posts\/309","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/comments?post=309"}],"version-history":[{"count":4,"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/posts\/309\/revisions"}],"predecessor-version":[{"id":335,"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/posts\/309\/revisions\/335"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/media\/686"}],"wp:attachment":[{"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/media?parent=309"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/categories?post=309"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/cwassignments.com\/blog\/wp-json\/wp\/v2\/tags?post=309"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}