{"id":1771,"date":"2019-08-09T14:04:13","date_gmt":"2019-08-09T18:04:13","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/?post_type=back-matter&p=1771"},"modified":"2019-08-09T15:15:55","modified_gmt":"2019-08-09T19:15:55","slug":"answer-key-10-4","status":"publish","type":"back-matter","link":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/back-matter\/answer-key-10-4\/","title":{"raw":"Answer Key 10.4","rendered":"Answer Key 10.4"},"content":{"raw":"[latexpage]\r\n
    \r\n \t
  1. \\(2^2-4(4)(-5)\\Rightarrow 4+80=84\\hspace{0.25in} \\therefore 2\\text{ real solutions}\\)<\/li>\r\n \t
  2. \\((-6)^2-4(9)(1)\\Rightarrow 36-36=0\\hspace{0.25in} \\therefore 1\\text{ real solution}\\)<\/li>\r\n \t
  3. \\((3)^2-4(2)(-5)\\Rightarrow 9+40=49\\hspace{0.25in} \\therefore 2\\text{ real solutions}\\)<\/li>\r\n \t
  4. \\(3x^2+5x-3\\Rightarrow (5)^2-4(3)(-3)\\Rightarrow 25+36=61\\hspace{0.25in} \\therefore 2\\text{ real solutions}\\)<\/li>\r\n \t
  5. \\(3x^2+5x-2\\Rightarrow (5)^2-4(3)(-2)\\Rightarrow 25+24=49\\hspace{0.25in} \\therefore 2\\text{ real solutions}\\)<\/li>\r\n \t
  6. \\((-8)^2-4(1)(16)\\Rightarrow 64-64=0\\hspace{0.25in} \\therefore 1\\text{ real solution}\\)<\/li>\r\n \t
  7. \\(a^2+10a-56\\Rightarrow (10)^2-4(1)(-56)\\Rightarrow 100+224=324\\hspace{0.25in} \\therefore 2\\text{ real solutions}\\)<\/li>\r\n \t
  8. \\(x^2-4x+4\\Rightarrow (-4)^2-4(1)(4)\\Rightarrow 16-16=0\\hspace{0.25in} \\therefore 1\\text{ real solution}\\)<\/li>\r\n \t
  9. \\(5x^2-10x+26\\Rightarrow (-10)^2-4(5)(26)\\Rightarrow 100-520=-420\\)\r\n\u2234 \\(2\\text{ non-real solutions}\\)<\/li>\r\n \t
  10. \\(n^2-10n+21\\Rightarrow (-10)^2-4(1)(21)\\Rightarrow 100-84=16\\)\r\n\u2234 \\(2\\text{ real solutions}\\)<\/li>\r\n<\/ol>\r\n \r\n
      \r\n \t
    1. \\(\\begin{array}{rrl}\r\n\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\\r\na&=&\\phantom{-}4 \\\\\r\nb&=&\\phantom{-}3 \\\\\r\nc&=&-6 \\\\ \\\\\r\na&=&\\dfrac{-3\\pm \\sqrt{3^2-4(4)(-6)}}{2(4)} \\\\ \\\\\r\na&=&\\dfrac{-3\\pm \\sqrt{9+96}}{8} \\\\ \\\\\r\na&=&\\dfrac{-3\\pm \\sqrt{105}}{8}\r\n\\end{array}\\)<\/li>\r\n \t
    2. \\(\\begin{array}{rrl}\r\n\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\\r\na&=&\\phantom{-}3 \\\\\r\nb&=&\\phantom{-}2 \\\\\r\nc&=&-3 \\\\ \\\\\r\nk&=&\\dfrac{-2\\pm \\sqrt{2^2-4(3)(-3)}}{2(3)} \\\\ \\\\\r\nk&=&\\dfrac{-2\\pm \\sqrt{4+36}}{6} \\\\ \\\\\r\nk&=&\\dfrac{-2\\pm \\sqrt{40}}{6} \\\\ \\\\\r\nk&=&\\dfrac{-2\\pm 2\\sqrt{10}}{6} \\Rightarrow \\dfrac{-1\\pm \\sqrt{10}}{3}\r\n\\end{array}\\)<\/li>\r\n \t
    3. \\(\\begin{array}{rrl}\r\n\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\\r\na&=&\\phantom{-}2 \\\\\r\nb&=&-8 \\\\\r\nc&=&-2 \\\\ \\\\\r\nx&=&\\dfrac{-(-8)\\pm \\sqrt{(-8)^2-4(2)(-2)}}{2(2)} \\\\ \\\\\r\nx&=&\\dfrac{8\\pm \\sqrt{64+16}}{4} \\\\ \\\\\r\nx&=&\\dfrac{8\\pm \\sqrt{80}}{4} \\\\ \\\\\r\nx&=&\\dfrac{8\\pm 4\\sqrt{5}}{4}\\Rightarrow 2\\pm \\sqrt{5}\r\n\\end{array}\\)<\/li>\r\n \t
    4. \\(\\begin{array}{rrl}\r\n\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\\r\na&=&\\phantom{-}6 \\\\\r\nb&=&\\phantom{-}8 \\\\\r\nc&=&-1 \\\\ \\\\\r\nn&=&\\dfrac{-8\\pm \\sqrt{8^2-4(6)(-1)}}{2(6)} \\\\ \\\\\r\nn&=&\\dfrac{-8\\pm \\sqrt{64+24}}{12} \\\\ \\\\\r\nn&=&\\dfrac{-8\\pm \\sqrt{88}}{12} \\\\ \\\\\r\nn&=&\\dfrac{-8\\pm 2\\sqrt{22}}{12}\\Rightarrow \\dfrac{-4\\pm \\sqrt{22}}{6}\r\n\\end{array}\\)<\/li>\r\n \t
    5. \\(\\begin{array}{rrl}\r\n\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\\r\na&=&\\phantom{-}2 \\\\\r\nb&=&-3 \\\\\r\nc&=&\\phantom{-}6 \\\\ \\\\\r\nm&=&\\dfrac{-(-3)\\pm \\sqrt{(-3)^2-4(2)(6)}}{2(2)} \\\\ \\\\\r\nm&=&\\dfrac{3\\pm \\sqrt{9-48}}{4} \\\\ \\\\\r\nm&=&\\dfrac{3\\pm \\sqrt{-39}}{4} \\\\ \\\\\r\n\\end{array}\\)\r\n\r\nA negative square root means there are 2 non-real solutions or no real solution.<\/li>\r\n \t
    6. \\(\\begin{array}{rrl}\r\n\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\\r\na&=&5 \\\\\r\nb&=&2 \\\\\r\nc&=&6 \\\\ \\\\\r\np&=&\\dfrac{-2\\pm \\sqrt{2^2-4(5)(6)}}{2(5)} \\\\ \\\\\r\np&=&\\dfrac{-2\\pm \\sqrt{4-120}}{10} \\\\ \\\\\r\np&=&\\dfrac{-2\\pm \\sqrt{-116}}{10}\r\n\\end{array}\\)\r\n\r\nA negative square root means there are 2 non-real solutions or no real solution.<\/li>\r\n \t
    7. \\(\\begin{array}{rrl}\r\n\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\\r\na&=&\\phantom{-}3 \\\\\r\nb&=&-2 \\\\\r\nc&=&-1 \\\\ \\\\\r\nr&=&\\dfrac{-(-2)\\pm \\sqrt{(-2)^2-4(3)(-1)}}{2(3)} \\\\ \\\\\r\nr&=&\\dfrac{2\\pm \\sqrt{4+12}}{6} \\\\ \\\\\r\nr&=&\\dfrac{2\\pm \\sqrt{16}}{6} \\\\ \\\\\r\nr&=&\\dfrac{2\\pm 4}{6} \\Rightarrow 1, -\\dfrac{1}{3}\r\n\\end{array}\\)<\/li>\r\n \t
    8. \\(\\begin{array}{rrl}\r\n\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\\r\na&=&\\phantom{-0}2 \\\\\r\nb&=&-\\phantom{0}2 \\\\\r\nc&=&-15 \\\\ \\\\\r\nx&=&\\dfrac{-(-2)\\pm \\sqrt{(-2)^2-4(2)(-15)}}{2(2)} \\\\ \\\\\r\nx&=&\\dfrac{2\\pm \\sqrt{4+120}}{4} \\\\ \\\\\r\nx&=&\\dfrac{2\\pm \\sqrt{124}}{4} \\\\ \\\\\r\nx&=&\\dfrac{2\\pm 2\\sqrt{31}}{4} \\Rightarrow \\dfrac{1\\pm \\sqrt{31}}{2}\r\n\\end{array}\\)<\/li>\r\n \t
    9. \\(\\begin{array}{rrl}\r\n\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\\r\na&=&\\phantom{0}4 \\\\\r\nb&=&-3 \\\\\r\nc&=&10 \\\\ \\\\\r\nn&=&\\dfrac{-(-3)\\pm \\sqrt{(-3)^2-4(4)(10)}}{2(4)} \\\\ \\\\\r\nn&=&\\dfrac{3\\pm \\sqrt{9-160}}{8} \\\\ \\\\\r\nn&=&\\dfrac{3\\pm \\sqrt{-151}}{8} \\\\ \\\\\r\n\\end{array}\\)\r\n\r\n\u2234 2 non-real solutions<\/li>\r\n \t
    10. \\(\\begin{array}{rrl}\r\n\\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\ \\\\\r\na&=&1 \\\\\r\nb&=&6 \\\\\r\nc&=&9 \\\\ \\\\\r\nb&=&\\dfrac{-6\\pm \\sqrt{6^2-4(1)(9)}}{2(1)} \\\\ \\\\\r\nb&=&\\dfrac{-6\\pm 0\\cancel{\\sqrt{36-36}}}{2} \\\\ \\\\\r\nb&=&\\dfrac{-6}{2}\\Rightarrow -3\r\n\\end{array}\\)<\/li>\r\n \t
    11. \\(\\begin{array}{rrrrrrrrrrr}\r\n\\\\ \\\\\r\nv^2&-&4v&-&5&=&-8&&&& \\\\\r\n&&&+&8&&+8&&&& \\\\\r\n\\midrule\r\n&&&&0&=&v^2&-&4v&+&3\r\n\\end{array}\\)\r\n\r\n\\(\\begin{array}{rrl}\r\na&=&\\phantom{-}1 \\\\\r\nb&=&-4 \\\\\r\nc&=&\\phantom{-}3 \\\\ \\\\\r\nv&=&\\dfrac{-(-4)\\pm \\sqrt{(-4)^2-4(1)(3)}}{2(1)} \\\\ \\\\\r\nv&=&\\dfrac{4\\pm \\sqrt{16-12}}{2} \\\\ \\\\\r\nv&=&\\dfrac{4\\pm \\sqrt{4}}{2} \\\\ \\\\\r\nv&=&\\dfrac{4\\pm 2}{2}\\Rightarrow 2 \\pm 1 \\\\ \\\\\r\nv&=&3, 1\r\n\\end{array}\\)<\/li>\r\n \t
    12. \\(\\begin{array}{rrrrrrrrrrr}\r\n\\\\ \\\\\r\nx^2&+&2x&+&6&=&4&&&& \\\\\r\n&&&-&4&&-4&&&& \\\\\r\n\\midrule\r\n&&&&0&=&x^2&+&2x&+&2\r\n\\end{array}\\)\r\n\r\n\\(\\begin{array}{rrl}\r\na&=&1 \\\\\r\nb&=&2 \\\\\r\nc&=&2 \\\\ \\\\\r\nx&=&\\dfrac{-2\\pm \\sqrt{2^2-4(1)(2)}}{2(1)} \\\\ \\\\\r\nx&=&\\dfrac{-2\\pm \\sqrt{4-8}}{2} \\\\ \\\\\r\nx&=&\\dfrac{-2\\pm \\sqrt{-4}}{2}\r\n\\end{array}\\)\r\n\r\n\u2234 2 non-real solutions<\/li>\r\n<\/ol>","rendered":"
        \n
      1. \"2^2-4(4)(-5)\Rightarrow 4+80=84\hspace{0.25in} \therefore 2\text{ real solutions}\"<\/li>\n
      2. \"(-6)^2-4(9)(1)\Rightarrow 36-36=0\hspace{0.25in} \therefore 1\text{ real solution}\"<\/li>\n
      3. \"(3)^2-4(2)(-5)\Rightarrow 9+40=49\hspace{0.25in} \therefore 2\text{ real solutions}\"<\/li>\n
      4. \"3x^2+5x-3\Rightarrow (5)^2-4(3)(-3)\Rightarrow 25+36=61\hspace{0.25in} \therefore 2\text{ real solutions}\"<\/li>\n
      5. \"3x^2+5x-2\Rightarrow (5)^2-4(3)(-2)\Rightarrow 25+24=49\hspace{0.25in} \therefore 2\text{ real solutions}\"<\/li>\n
      6. \"(-8)^2-4(1)(16)\Rightarrow 64-64=0\hspace{0.25in} \therefore 1\text{ real solution}\"<\/li>\n
      7. \"a^2+10a-56\Rightarrow (10)^2-4(1)(-56)\Rightarrow 100+224=324\hspace{0.25in} \therefore 2\text{ real solutions}\"<\/li>\n
      8. \"x^2-4x+4\Rightarrow (-4)^2-4(1)(4)\Rightarrow 16-16=0\hspace{0.25in} \therefore 1\text{ real solution}\"<\/li>\n
      9. \"5x^2-10x+26\Rightarrow (-10)^2-4(5)(26)\Rightarrow 100-520=-420\"
        \n\u2234 \"2\text{ non-real solutions}\"<\/li>\n
      10. \"n^2-10n+21\Rightarrow (-10)^2-4(1)(21)\Rightarrow 100-84=16\"
        \n\u2234 \"2\text{ real solutions}\"<\/li>\n<\/ol>\n

         <\/p>\n

          \n
        1. \"\begin{array}{rrl}<\/li>\n
        2. \"\begin{array}{rrl}<\/li>\n
        3. \"\begin{array}{rrl}<\/li>\n
        4. \"\begin{array}{rrl}<\/li>\n
        5. \"\begin{array}{rrl}\n

          A negative square root means there are 2 non-real solutions or no real solution.<\/li>\n

        6. \"\begin{array}{rrl}\n

          A negative square root means there are 2 non-real solutions or no real solution.<\/li>\n

        7. \"\begin{array}{rrl}<\/li>\n
        8. \"\begin{array}{rrl}<\/li>\n
        9. \"\begin{array}{rrl}\n

          \u2234 2 non-real solutions<\/li>\n

        10. \"\begin{array}{rrl}<\/li>\n
        11. \"\begin{array}{rrrrrrrrrrr}\n

          \"\begin{array}{rrl}<\/li>\n

        12. \"\begin{array}{rrrrrrrrrrr}\n

          \"\begin{array}{rrl}<\/p>\n

          \u2234 2 non-real solutions<\/li>\n<\/ol>\n","protected":false},"author":727,"menu_order":91,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"back-matter-type":[],"contributor":[],"license":[],"class_list":["post-1771","back-matter","type-back-matter","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter\/1771","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/types\/back-matter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/users\/727"}],"version-history":[{"count":3,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter\/1771\/revisions"}],"predecessor-version":[{"id":1774,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter\/1771\/revisions\/1774"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter\/1771\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/media?parent=1771"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter-type?post=1771"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/contributor?post=1771"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/license?post=1771"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}