{"id":1984,"date":"2021-12-02T19:40:19","date_gmt":"2021-12-03T00:40:19","guid":{"rendered":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-10-5\/"},"modified":"2022-11-02T10:38:55","modified_gmt":"2022-11-02T14:38:55","slug":"answer-key-10-5","status":"publish","type":"back-matter","link":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/back-matter\/answer-key-10-5\/","title":{"raw":"Answer Key 10.5","rendered":"Answer Key 10.5"},"content":{"raw":"
    \n \t
  1. [latex]\\text{let }u=x^2 [\/latex]\n[latex]\\therefore u^2-5u+4=0 [\/latex]\n[latex]\\text{factors to }(u-4)(u-1)=0 [\/latex]\n[latex]\\text{replace }u: (x^2-4)(x^2-1)=0 [\/latex]\n[latex](x-2)(x+2)(x-1)(x+1)=0 [\/latex]\n[latex]x=\\pm 2, \\pm 1[\/latex]<\/li>\n \t
  2. [latex]\\text{let }u=y^2[\/latex]\n[latex]\\therefore u^2-9y+20=0[\/latex]\n[latex]\\text{factors to }(u-5)(u-4)=0[\/latex]\n[latex]\\text{replace }u: (y^2-5)(y^2-4)=0 [\/latex]\n[latex]\\begin{array}{ll}\ny^2-5=0\\hspace{0.25in}&(y-2)(y+2)=0 \\\\\ny^2=5&y=\\pm 2 \\\\\ny=\\pm \\sqrt{5}&\n\\end{array}[\/latex]<\/li>\n \t
  3. [latex]u=m^2[\/latex]\n[latex]\\therefore u^2-7u-8=0[\/latex]\n[latex](u-8)(u+1)=0[\/latex]\n[latex](m^2-8)(m^2+1)=0[\/latex]\n[latex](m+\\sqrt{8})(m-\\sqrt{8})(m^2+1)=0 [\/latex]\n[latex]m=\\pm \\sqrt{8}\\text{ or }\\pm 2\\sqrt{2}[\/latex]\n[latex]m^2+1\\text{ has 2 non-real solutions}[\/latex]<\/li>\n \t
  4. [latex]u=y^2[\/latex]\n[latex]\\therefore u^2-29y+100=0[\/latex]\n[latex](u-25)(u-4)=0[\/latex]\n[latex](y^2-25)(y^2-4)=0[\/latex]\n[latex](y-5)(y+5)(y-2)(y+2)=0[\/latex]\n[latex]y=\\pm 5, \\pm 2[\/latex]<\/li>\n \t
  5. [latex]\\text{let }u=a^2[\/latex]\n[latex]\\therefore u^2-50u+49=0[\/latex]\n[latex](u-49)(u-1)=0[\/latex]\n[latex](a^2-49)(a^2-1)=0 [\/latex]\n[latex](a-7)(a+7)(a-1)(a+1)=0[\/latex]\n[latex]a=\\pm 7, \\pm1[\/latex]<\/li>\n \t
  6. [latex]\\text{let }u=b^2[\/latex]\n[latex]\\therefore u^2-10u+9=0[\/latex]\n[latex](u-9)(u-1)=0[\/latex]\n[latex](b^2-9)(b^2-1)=0[\/latex]\n[latex](b-3)(b+3)(b-1)(b+1)=0[\/latex]\n[latex]b=\\pm 3, \\pm 1[\/latex]<\/li>\n \t
  7. [latex]x^4-20x^2+64=0[\/latex]\n[latex]\\text{let }u=x^2[\/latex]\n[latex]\\therefore u^2-20u+64=0[\/latex]\n[latex](u-16)(u-4)=0[\/latex]\n[latex](x^2-16)(x^2-4)=0[\/latex]\n[latex](x-4)(x+4)(x-2)(x+2)=0[\/latex]\n[latex]x=\\pm 4, \\pm 2[\/latex]<\/li>\n \t
  8. [latex]6z^6-z^3-12=0[\/latex]\n[latex]\\text{let }u=z^3[\/latex]\n[latex]\\therefore 6u^2-u-12=0[\/latex]\n[latex](3u+4)(2u-3)=0[\/latex]\n[latex](3z^3+4)(2z^3-3)=0[\/latex]\n[latex]\\begin{array}{ll}\n3z^3+4=0\\hspace{0.25in}&2z^3-3=0 \\\\\n3z^3=-4&2z^3=3 \\\\ \\\\\nz^3=-\\dfrac{4}{3}&z^3=\\dfrac{3}{2} \\\\ \\\\\nz=\\sqrt[3]{-\\dfrac{4}{3}}&z=\\sqrt[3]{\\dfrac{3}{2}}\n\\end{array}[\/latex]<\/li>\n \t
  9. [latex]z^6-19z^3-216=0[\/latex]\n[latex]\\text{let }u=z^3[\/latex]\n[latex]\\therefore u^2-19u-216=0[\/latex]\n[latex](u-27)(u+8)=0[\/latex]\n[latex](z^3-27)(z^3+8)=0[\/latex]\n[latex](z-3)(z^2+3z+9)(z+2)(z^2-2z+4)=0[\/latex]\n[latex]z=3, -2[\/latex]\n[latex]2\\text{ non-real solutions each for the 2nd and 4th factors}[\/latex]<\/li>\n \t
  10. [latex]\\text{let }u=x^3[\/latex]\n[latex]\\therefore u^2-35u+216=0[\/latex]\n[latex](u-27)(u-8)=0[\/latex]\n[latex](x^3-27)(x^3-8)=0[\/latex]\n[latex](x-3)(x^2+3x+9)(x-2)(x^2+2x+4)[\/latex]\n[latex]x=2, 3[\/latex]\n[latex]2\\text{ non-real solutions each for the 2nd and 4th factors}[\/latex]<\/li>\n<\/ol>","rendered":"
      \n
    1. [latex]\\text{let }u=x^2[\/latex]
      \n[latex]\\therefore u^2-5u+4=0[\/latex]
      \n[latex]\\text{factors to }(u-4)(u-1)=0[\/latex]
      \n[latex]\\text{replace }u: (x^2-4)(x^2-1)=0[\/latex]
      \n[latex](x-2)(x+2)(x-1)(x+1)=0[\/latex]
      \n[latex]x=\\pm 2, \\pm 1[\/latex]<\/li>\n
    2. [latex]\\text{let }u=y^2[\/latex]
      \n[latex]\\therefore u^2-9y+20=0[\/latex]
      \n[latex]\\text{factors to }(u-5)(u-4)=0[\/latex]
      \n[latex]\\text{replace }u: (y^2-5)(y^2-4)=0[\/latex]
      \n[latex]\\begin{array}{ll} y^2-5=0\\hspace{0.25in}&(y-2)(y+2)=0 \\\\ y^2=5&y=\\pm 2 \\\\ y=\\pm \\sqrt{5}& \\end{array}[\/latex]<\/li>\n
    3. [latex]u=m^2[\/latex]
      \n[latex]\\therefore u^2-7u-8=0[\/latex]
      \n[latex](u-8)(u+1)=0[\/latex]
      \n[latex](m^2-8)(m^2+1)=0[\/latex]
      \n[latex](m+\\sqrt{8})(m-\\sqrt{8})(m^2+1)=0[\/latex]
      \n[latex]m=\\pm \\sqrt{8}\\text{ or }\\pm 2\\sqrt{2}[\/latex]
      \n[latex]m^2+1\\text{ has 2 non-real solutions}[\/latex]<\/li>\n
    4. [latex]u=y^2[\/latex]
      \n[latex]\\therefore u^2-29y+100=0[\/latex]
      \n[latex](u-25)(u-4)=0[\/latex]
      \n[latex](y^2-25)(y^2-4)=0[\/latex]
      \n[latex](y-5)(y+5)(y-2)(y+2)=0[\/latex]
      \n[latex]y=\\pm 5, \\pm 2[\/latex]<\/li>\n
    5. [latex]\\text{let }u=a^2[\/latex]
      \n[latex]\\therefore u^2-50u+49=0[\/latex]
      \n[latex](u-49)(u-1)=0[\/latex]
      \n[latex](a^2-49)(a^2-1)=0[\/latex]
      \n[latex](a-7)(a+7)(a-1)(a+1)=0[\/latex]
      \n[latex]a=\\pm 7, \\pm1[\/latex]<\/li>\n
    6. [latex]\\text{let }u=b^2[\/latex]
      \n[latex]\\therefore u^2-10u+9=0[\/latex]
      \n[latex](u-9)(u-1)=0[\/latex]
      \n[latex](b^2-9)(b^2-1)=0[\/latex]
      \n[latex](b-3)(b+3)(b-1)(b+1)=0[\/latex]
      \n[latex]b=\\pm 3, \\pm 1[\/latex]<\/li>\n
    7. [latex]x^4-20x^2+64=0[\/latex]
      \n[latex]\\text{let }u=x^2[\/latex]
      \n[latex]\\therefore u^2-20u+64=0[\/latex]
      \n[latex](u-16)(u-4)=0[\/latex]
      \n[latex](x^2-16)(x^2-4)=0[\/latex]
      \n[latex](x-4)(x+4)(x-2)(x+2)=0[\/latex]
      \n[latex]x=\\pm 4, \\pm 2[\/latex]<\/li>\n
    8. [latex]6z^6-z^3-12=0[\/latex]
      \n[latex]\\text{let }u=z^3[\/latex]
      \n[latex]\\therefore 6u^2-u-12=0[\/latex]
      \n[latex](3u+4)(2u-3)=0[\/latex]
      \n[latex](3z^3+4)(2z^3-3)=0[\/latex]
      \n[latex]\\begin{array}{ll} 3z^3+4=0\\hspace{0.25in}&2z^3-3=0 \\\\ 3z^3=-4&2z^3=3 \\\\ \\\\ z^3=-\\dfrac{4}{3}&z^3=\\dfrac{3}{2} \\\\ \\\\ z=\\sqrt[3]{-\\dfrac{4}{3}}&z=\\sqrt[3]{\\dfrac{3}{2}} \\end{array}[\/latex]<\/li>\n
    9. [latex]z^6-19z^3-216=0[\/latex]
      \n[latex]\\text{let }u=z^3[\/latex]
      \n[latex]\\therefore u^2-19u-216=0[\/latex]
      \n[latex](u-27)(u+8)=0[\/latex]
      \n[latex](z^3-27)(z^3+8)=0[\/latex]
      \n[latex](z-3)(z^2+3z+9)(z+2)(z^2-2z+4)=0[\/latex]
      \n[latex]z=3, -2[\/latex]
      \n[latex]2\\text{ non-real solutions each for the 2nd and 4th factors}[\/latex]<\/li>\n
    10. [latex]\\text{let }u=x^3[\/latex]
      \n[latex]\\therefore u^2-35u+216=0[\/latex]
      \n[latex](u-27)(u-8)=0[\/latex]
      \n[latex](x^3-27)(x^3-8)=0[\/latex]
      \n[latex](x-3)(x^2+3x+9)(x-2)(x^2+2x+4)[\/latex]
      \n[latex]x=2, 3[\/latex]
      \n[latex]2\\text{ non-real solutions each for the 2nd and 4th factors}[\/latex]<\/li>\n<\/ol>\n","protected":false},"author":90,"menu_order":97,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by-nc-sa"},"back-matter-type":[],"contributor":[],"license":[56],"class_list":["post-1984","back-matter","type-back-matter","status-publish","hentry","license-cc-by-nc-sa"],"_links":{"self":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1984","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter"}],"about":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/types\/back-matter"}],"author":[{"embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/users\/90"}],"version-history":[{"count":1,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1984\/revisions"}],"predecessor-version":[{"id":1985,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1984\/revisions\/1985"}],"metadata":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter\/1984\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/media?parent=1984"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/pressbooks\/v2\/back-matter-type?post=1984"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/contributor?post=1984"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/opentextbc.ca\/intermediatealgebraberg\/wp-json\/wp\/v2\/license?post=1984"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}