{"id":1777,"date":"2019-08-09T16:56:06","date_gmt":"2019-08-09T20:56:06","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/?post_type=back-matter&p=1777"},"modified":"2019-08-09T18:02:00","modified_gmt":"2019-08-09T22:02:00","slug":"answer-key-10-5","status":"publish","type":"back-matter","link":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/back-matter\/answer-key-10-5\/","title":{"raw":"Answer Key 10.5","rendered":"Answer Key 10.5"},"content":{"raw":"[latexpage]\r\n
    \r\n \t
  1. \\(\\text{let }u=x^2 \\)\r\n\\(\\therefore u^2-5u+4=0 \\)\r\n\\(\\text{factors to }(u-4)(u-1)=0 \\)\r\n\\(\\text{replace }u: (x^2-4)(x^2-1)=0 \\\\ \\)\r\n\\((x-2)(x+2)(x-1)(x+1)=0 \\)\r\n\\(x=\\pm 2, \\pm 1\\)<\/li>\r\n \t
  2. \\(\\text{let }u=y^2\\)\r\n\\(\\therefore u^2-9y+20=0\\)\r\n\\(\\text{factors to }(u-5)(u-4)=0\\)\r\n\\(\\text{replace }u: (y^2-5)(y^2-4)=0 \\\\ \\)\r\n\\(\\begin{array}{ll}\r\ny^2-5=0\\hspace{0.25in}&(y-2)(y+2)=0 \\\\\r\ny^2=5&y=\\pm 2 \\\\\r\ny=\\pm \\sqrt{5}&\r\n\\end{array}\\)<\/li>\r\n \t
  3. \\(u=m^2\\)\r\n\\(\\therefore u^2-7u-8=0\\)\r\n\\((u-8)(u+1)=0\\)\r\n\\((m^2-8)(m^2+1)=0 \\\\ \\)\r\n\\((m+\\sqrt{8})(m-\\sqrt{8})(m^2+1)=0 \\)\r\n\\(m=\\pm \\sqrt{8}\\text{ or }\\pm 2\\sqrt{2}\\)\r\n\\(m^2+1\\text{ has 2 non-real solutions}\\)<\/li>\r\n \t
  4. \\(u=y^2\\)\r\n\\(\\therefore u^2-29y+100=0\\)\r\n\\((u-25)(u-4)=0\\)\r\n\\((y^2-25)(y^2-4)=0 \\\\ \\)\r\n\\((y-5)(y+5)(y-2)(y+2)=0\\)\r\n\\(y=\\pm 5, \\pm 2\\)<\/li>\r\n \t
  5. \\(\\text{let }u=a^2\\)\r\n\\(\\therefore u^2-50u+49=0\\)\r\n\\((u-49)(u-1)=0\\)\r\n\\((a^2-49)(a^2-1)=0 \\\\ \\)\r\n\\((a-7)(a+7)(a-1)(a+1)=0\\)\r\n\\(a=\\pm 7, \\pm1\\)<\/li>\r\n \t
  6. \\(\\text{let }u=b^2\\)\r\n\\(\\therefore u^2-10u+9=0\\)\r\n\\((u-9)(u-1)=0\\)\r\n\\((b^2-9)(b^2-1)=0 \\\\ \\)\r\n\\((b-3)(b+3)(b-1)(b+1)=0\\)\r\n\\(b=\\pm 3, \\pm 1\\)<\/li>\r\n \t
  7. \\(x^4-20x^2+64=0\\)\r\n\\(\\text{let }u=x^2\\)\r\n\\(\\therefore u^2-20u+64=0\\)\r\n\\((u-16)(u-4)=0\\)\r\n\\((x^2-16)(x^2-4)=0 \\\\ \\)\r\n\\((x-4)(x+4)(x-2)(x+2)=0\\)\r\n\\(x=\\pm 4, \\pm 2\\)<\/li>\r\n \t
  8. \\(6z^6-z^3-12=0\\)\r\n\\(\\text{let }u=z^3\\)\r\n\\(\\therefore 6u^2-u-12=0\\)\r\n\\((3u+4)(2u-3)=0\\)\r\n\\((3z^3+4)(2z^3-3)=0\\\\ \\)\r\n\\(\\begin{array}{ll}\r\n3z^3+4=0\\hspace{0.25in}&2z^3-3=0 \\\\\r\n3z^3=-4&2z^3=3 \\\\ \\\\\r\nz^3=-\\dfrac{4}{3}&z^3=\\dfrac{3}{2} \\\\ \\\\\r\nz=\\sqrt[3]{-\\dfrac{4}{3}}&z=\\sqrt[3]{\\dfrac{3}{2}}\r\n\\end{array}\\)<\/li>\r\n \t
  9. \\(z^6-19z^3-216=0\\)\r\n\\(\\text{let }u=z^3\\)\r\n\\(\\therefore u^2-19u-216=0\\)\r\n\\((u-27)(u+8)=0\\)\r\n\\((z^3-27)(z^3+8)=0 \\\\ \\)\r\n\\((z-3)(z^2+3z+9)(z+2)(z^2-2z+4)=0\\)\r\n\\(z=3, -2\\)\r\n\\(2\\text{ non-real solutions each for the 2nd and 4th factors}\\)<\/li>\r\n \t
  10. \\(\\text{let }u=x^3\\)\r\n\\(\\therefore u^2-35u+216=0\\)\r\n\\((u-27)(u-8)=0\\)\r\n\\((x^3-27)(x^3-8)=0\\)\r\n\\((x-3)(x^2+3x+9)(x-2)(x^2+2x+4)\\)\r\n\\(x=2, 3\\)\r\n\\(2\\text{ non-real solutions each for the 2nd and 4th factors}\\)<\/li>\r\n<\/ol>","rendered":"
      \n
    1. \"\text{let }u=x^2\"
      \n\"\therefore u^2-5u+4=0\"
      \n\"\text{factors to }(u-4)(u-1)=0\"
      \n\"\text{replace }u: (x^2-4)(x^2-1)=0 \\\"
      \n\"(x-2)(x+2)(x-1)(x+1)=0\"
      \n\"x=\pm 2, \pm 1\"<\/li>\n
    2. \"\text{let }u=y^2\"
      \n\"\therefore u^2-9y+20=0\"
      \n\"\text{factors to }(u-5)(u-4)=0\"
      \n\"\text{replace }u: (y^2-5)(y^2-4)=0 \\\"
      \n\"\begin{array}{ll}<\/li>\n
    3. \"u=m^2\"
      \n\"\therefore u^2-7u-8=0\"
      \n\"(u-8)(u+1)=0\"
      \n\"(m^2-8)(m^2+1)=0 \\\"
      \n\"(m+\sqrt{8})(m-\sqrt{8})(m^2+1)=0\"
      \n\"m=\pm \sqrt{8}\text{ or }\pm 2\sqrt{2}\"
      \n\"m^2+1\text{ has 2 non-real solutions}\"<\/li>\n
    4. \"u=y^2\"
      \n\"\therefore u^2-29y+100=0\"
      \n\"(u-25)(u-4)=0\"
      \n\"(y^2-25)(y^2-4)=0 \\\"
      \n\"(y-5)(y+5)(y-2)(y+2)=0\"
      \n\"y=\pm 5, \pm 2\"<\/li>\n
    5. \"\text{let }u=a^2\"
      \n\"\therefore u^2-50u+49=0\"
      \n\"(u-49)(u-1)=0\"
      \n\"(a^2-49)(a^2-1)=0 \\\"
      \n\"(a-7)(a+7)(a-1)(a+1)=0\"
      \n\"a=\pm 7, \pm1\"<\/li>\n
    6. \"\text{let }u=b^2\"
      \n\"\therefore u^2-10u+9=0\"
      \n\"(u-9)(u-1)=0\"
      \n\"(b^2-9)(b^2-1)=0 \\\"
      \n\"(b-3)(b+3)(b-1)(b+1)=0\"
      \n\"b=\pm 3, \pm 1\"<\/li>\n
    7. \"x^4-20x^2+64=0\"
      \n\"\text{let }u=x^2\"
      \n\"\therefore u^2-20u+64=0\"
      \n\"(u-16)(u-4)=0\"
      \n\"(x^2-16)(x^2-4)=0 \\\"
      \n\"(x-4)(x+4)(x-2)(x+2)=0\"
      \n\"x=\pm 4, \pm 2\"<\/li>\n
    8. \"6z^6-z^3-12=0\"
      \n\"\text{let }u=z^3\"
      \n\"\therefore 6u^2-u-12=0\"
      \n\"(3u+4)(2u-3)=0\"
      \n\"(3z^3+4)(2z^3-3)=0\\\"
      \n\"\begin{array}{ll}<\/li>\n
    9. \"z^6-19z^3-216=0\"
      \n\"\text{let }u=z^3\"
      \n\"\therefore u^2-19u-216=0\"
      \n\"(u-27)(u+8)=0\"
      \n\"(z^3-27)(z^3+8)=0 \\\"
      \n\"(z-3)(z^2+3z+9)(z+2)(z^2-2z+4)=0\"
      \n\"z=3, -2\"
      \n\"2\text{ non-real solutions each for the 2nd and 4th factors}\"<\/li>\n
    10. \"\text{let }u=x^3\"
      \n\"\therefore u^2-35u+216=0\"
      \n\"(u-27)(u-8)=0\"
      \n\"(x^3-27)(x^3-8)=0\"
      \n\"(x-3)(x^2+3x+9)(x-2)(x^2+2x+4)\"
      \n\"x=2, 3\"
      \n\"2\text{ non-real solutions each for the 2nd and 4th factors}\"<\/li>\n<\/ol>\n","protected":false},"author":727,"menu_order":92,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"back-matter-type":[],"contributor":[],"license":[],"class_list":["post-1777","back-matter","type-back-matter","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter\/1777","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\/1777\/revisions"}],"predecessor-version":[{"id":1780,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter\/1777\/revisions\/1780"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter\/1777\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/media?parent=1777"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/pressbooks\/v2\/back-matter-type?post=1777"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/contributor?post=1777"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/intermediatealgebrakpu\/wp-json\/wp\/v2\/license?post=1777"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}