Answer Key 10.5

  1. \text{let }u=x^2
    \therefore u^2-5u+4=0
    \text{factors to }(u-4)(u-1)=0
    \text{replace }u: (x^2-4)(x^2-1)=0 \\
    (x-2)(x+2)(x-1)(x+1)=0
    x=\pm 2, \pm 1
  2. \text{let }u=y^2
    \therefore u^2-9y+20=0
    \text{factors to }(u-5)(u-4)=0
    \text{replace }u: (y^2-5)(y^2-4)=0 \\
    \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}
  3. u=m^2
    \therefore u^2-7u-8=0
    (u-8)(u+1)=0
    (m^2-8)(m^2+1)=0 \\
    (m+\sqrt{8})(m-\sqrt{8})(m^2+1)=0
    m=\pm \sqrt{8}\text{ or }\pm 2\sqrt{2}
    m^2+1\text{ has 2 non-real solutions}
  4. u=y^2
    \therefore u^2-29y+100=0
    (u-25)(u-4)=0
    (y^2-25)(y^2-4)=0 \\
    (y-5)(y+5)(y-2)(y+2)=0
    y=\pm 5, \pm 2
  5. \text{let }u=a^2
    \therefore u^2-50u+49=0
    (u-49)(u-1)=0
    (a^2-49)(a^2-1)=0 \\
    (a-7)(a+7)(a-1)(a+1)=0
    a=\pm 7, \pm1
  6. \text{let }u=b^2
    \therefore u^2-10u+9=0
    (u-9)(u-1)=0
    (b^2-9)(b^2-1)=0 \\
    (b-3)(b+3)(b-1)(b+1)=0
    b=\pm 3, \pm 1
  7. x^4-20x^2+64=0
    \text{let }u=x^2
    \therefore u^2-20u+64=0
    (u-16)(u-4)=0
    (x^2-16)(x^2-4)=0 \\
    (x-4)(x+4)(x-2)(x+2)=0
    x=\pm 4, \pm 2
  8. 6z^6-z^3-12=0
    \text{let }u=z^3
    \therefore 6u^2-u-12=0
    (3u+4)(2u-3)=0
    (3z^3+4)(2z^3-3)=0\\
    \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}
  9. z^6-19z^3-216=0
    \text{let }u=z^3
    \therefore u^2-19u-216=0
    (u-27)(u+8)=0
    (z^3-27)(z^3+8)=0 \\
    (z-3)(z^2+3z+9)(z+2)(z^2-2z+4)=0
    z=3, -2
    2\text{ non-real solutions each for the 2nd and 4th factors}
  10. \text{let }u=x^3
    \therefore u^2-35u+216=0
    (u-27)(u-8)=0
    (x^3-27)(x^3-8)=0
    (x-3)(x^2+3x+9)(x-2)(x^2+2x+4)
    x=2, 3
    2\text{ non-real solutions each for the 2nd and 4th factors}

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Intermediate Algebra by Terrance Berg is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

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