image imagewidth (px) 2 3.75k | material_type stringclasses 2
values | latex listlengths 1 1 |
|---|---|---|
HW | [
"$\\displaystyle x^{2}=\\left(-\\frac{1}{2}-\\frac{\\sqrt{3} i}{2}\\right)^{2}=\\left(\\frac{1}{2}+\\frac{\\sqrt{3}i}{2}\\right)^{2}=\\frac{1}{4}+\\frac{\\sqrt{3}}{2}i-\\frac{3}{2}=\\frac{5}{4}+\\frac{\\sqrt{3}}{2}i$"
] | |
HW | [
"$\\displaystyle =-5+2 \\sqrt{3} i-2-2 \\sqrt{5}i=3 \\quad \\therefore 3$"
] | |
HW | [
"3"
] | |
HW | [
"3"
] | |
HW | [
"97. $\\displaystyle x=-7+8 i$"
] | |
HW | [
"97. $\\displaystyle x=-7+8 i$"
] | |
HW | [
"$\\displaystyle x+7=-8 i,(x+7)^{2}=-64$"
] | |
HW | [
"$\\displaystyle x+7=-8 i,(x+7)^{2}=-64$"
] | |
HW | [
"$\\displaystyle x^{2}-2 x+1=0 $"
] | |
HW | [
"$\\displaystyle \\alpha+\\frac{1}{\\alpha}=2 .$"
] | |
HW | [
"$\\displaystyle \\alpha^{2}+\\frac{1}{\\alpha^{2}}=\\left(\\alpha+\\frac{1}{\\alpha}\\right)^{2}-2=2^{2}-2=2 .$"
] | |
HW | [
"$\\displaystyle \\alpha^{2}+\\frac{1}{\\alpha^{2}}=\\left(\\alpha+\\frac{1}{\\alpha}\\right)^{2}-2=2^{2}-2=2 .$"
] | |
HW | [
"$\\displaystyle 2$"
] | |
HW | [
"$\\displaystyle 2$"
] | |
HW | [
"$\\displaystyle \\frac{1}{\\alpha}+\\frac{1}{\\beta}=\\frac{\\alpha+\\beta}{\\alpha \\beta}=-\\frac{1}{11}$"
] | |
HW | [
"$\\displaystyle \\frac{1}{\\alpha}+\\frac{1}{\\delta}=\\frac{\\alpha+\\delta}{\\alpha\\delta}=\\frac{1}{11}$"
] | |
HW | [
"$\\displaystyle 0$"
] | |
HW | [
"$\\displaystyle (\\sqrt{\\alpha}+\\sqrt{\\beta})^{2}=2\\sqrt{\\alpha \\beta}+\\alpha+\\beta$"
] | |
HW | [
"$\\displaystyle \\frac{19}{2}$"
] | |
HW | [
"$\\displaystyle \\frac{19}{2}$"
] | |
HW | [
"$\\displaystyle 1-2 k+1<0, \\quad 2<2 k, k k$"
] | |
HW | [
"$\\displaystyle 1-2 k+1<0, \\quad 2<2 k, k k$"
] | |
HW | [
"$\\displaystyle \\frac{4 k^{2}-4 k+1}{4}-\\frac{24 k-23}{4}=0$"
] | |
HW | [
"$\\displaystyle k=1$ or $\\displaystyle 6$"
] | |
HW | [
"$\\displaystyle k=1$ or $\\displaystyle 6$"
] | |
HW | [
"$\\displaystyle 1 < k \\rightarrow k=6$"
] | |
HW | [
"$\\displaystyle 6$"
] | |
HW | [
"$\\displaystyle 6$"
] | |
HW | [
"$\\displaystyle 2 5{a}^{2} k^{2}+b^{2}-4 k^{2}-4 c+4=0$"
] | |
HW | [
"$\\displaystyle =0$"
] | |
HW | [
"$\\displaystyle a^{2}=\\frac{4}{25} \\quad c^{2}=1 \\quad b^{2}=0$"
] | |
HW | [
"$\\displaystyle a^{2}=\\frac{4}{25} \\quad c^{2}=1 \\quad b^{2}=0$"
] | |
HW | [
"$\\displaystyle \\frac{4}{25}+\\frac{25}{28}+0=\\frac{29}{25}$"
] | |
HW | [
" $\\displaystyle 129. \\; 4{a}^{2}-4({b}^{2}-9b+18)=0$ "
] | |
HW | [
"$\\displaystyle (a+b)(a-b)+a(b-2)=0$"
] | |
HW | [
"$\\displaystyle (a+b)(a-b)+a(b-2)=0$"
] | |
HW | [
"$\\displaystyle x<\\frac{2}{3} \\quad x \\geq \\frac{2}{3}$"
] | |
HW | [
"$\\displaystyle x<\\frac{2}{3} \\quad x \\geq \\frac{2}{3}$"
] | |
HW | [
"$\\displaystyle x^{2}-3 x+2=4 \\quad x^{2}+3 x-2-4=0$"
] | |
HW | [
"$\\displaystyle x^{2}-3x-2=0 \\quad x^{2}+3x-8=0$"
] | |
HW | [
"$\\displaystyle x^{2}-3x-2=0 \\quad x^{2}+3x-8=0$"
] | |
HW | [
"$\\displaystyle x=\\frac{3 \\pm \\sqrt{17}}{2} \\quad x=\\frac{-3 \\pm \\sqrt{41}}{2}$"
] | |
HW | [
"$\\displaystyle \\frac{3-\\sqrt{17}}{2}+\\frac{-3+\\sqrt{41}}{2}=\\frac{\\sqrt{41}}{2}-\\frac{\\sqrt{17}}{2}$"
] | |
HW | [
"$\\displaystyle \\frac{3-\\sqrt{17}}{2}+\\frac{-3+\\sqrt{41}}{2}=\\frac{\\sqrt{41}}{2}-\\frac{\\sqrt{17}}{2}$"
] | |
HW | [
"143. $\\displaystyle \\sqrt{5} \\sqrt{-245}$"
] | |
HW | [
"143. $\\displaystyle \\sqrt{5} \\sqrt{-245}$"
] | |
HW | [
"$\\displaystyle =\\sqrt{5} \\sqrt{245}i$"
] | |
HW | [
"$\\displaystyle =\\sqrt{5} \\sqrt{245}i$"
] | |
HW | [
"$\\displaystyle =\\sqrt{7^{2} \\times 5^{2}}i$"
] | |
HW | [
"$\\displaystyle =\\sqrt{7^{2} \\times 5^{2}}i$"
] | |
HW | [
"$\\displaystyle =35 i$"
] | |
HW | [
"$\\displaystyle =35 i$"
] | |
HW | [
"$\\displaystyle \\therefore 35i$"
] | |
HW | [
"good."
] | |
HW | [
"good."
] | |
HW | [
"$\\displaystyle \\alpha^{3}+\\beta^{3}=(\\alpha+\\beta)\\left(\\alpha^{2}-\\alpha \\beta+\\beta^{2}\\right)$"
] | |
HW | [
"$\\displaystyle \\alpha^{3}+\\beta^{3}=(\\alpha+\\beta)\\left(\\alpha^{2}-\\alpha \\beta+\\beta^{2}\\right)$"
] | |
HW | [
"$\\displaystyle =(\\alpha+\\beta)\\left\\{(\\alpha+\\beta)^{2}-3 \\alpha \\beta\\right\\}$"
] | |
HW | [
"720"
] | |
HW | [
"720"
] | |
HW | [
"$\\displaystyle x-2+28=x^{2}+x+14$"
] | |
HW | [
"$\\displaystyle x=2 \\sqrt{3} .$"
] | |
HW | [
"$\\displaystyle x=2 \\sqrt{3} .$"
] | |
HW | [
"$\\displaystyle -x+2+28=x^{2}+x+14$"
] | |
HW | [
"$\\displaystyle -x+2+28=x^{2}+x+14$"
] | |
HW | [
"$\\displaystyle 2 \\sqrt{2} i \\cdot 8 \\sqrt{2} i=-32$"
] | |
HW | [
"20"
] | |
HW | [
"20"
] | |
HW | [
"$\\displaystyle (2-i)(a+b i)+4 i(a-b i)=3-i .$"
] | |
HW | [
"$\\displaystyle (2-i)(a+b i)+4 i(a-b i)=3-i .$"
] | |
HW | [
"$\\displaystyle (2 a+b)+(2 b-a) i+4 a i+4 b=3-i$"
] | |
HW | [
"$\\displaystyle (2 a+b)+(2 b-a) i+4 a i+4 b=3-i$"
] | |
HW | [
"$\\displaystyle 2 a+5 b=3$"
] | |
HW | [
"$\\displaystyle 2 a+5 b=3$"
] | |
HW | [
"$\\displaystyle -1+i$"
] | |
HW | [
"$\\displaystyle -1+i$"
] | |
HW | [
" $\\displaystyle 125. \\; 36\\sqrt{7}i=0+36\\sqrt{7}i$"
] | |
HW | [
" $\\displaystyle 125. \\; 36\\sqrt{7}i=0+36\\sqrt{7}i$"
] | |
HW | [
"$\\displaystyle =\\left(i^{4}\\right)^{4}-\\left(i^{4}\\right)^{4} i+(i^{4})^{4} \\cdot i^{2}-\\left(i^{4}\\right)^{5} \\cdot i^{3}$"
] | |
HW | [
"$\\displaystyle =\\left(i^{4}\\right)^{4}-\\left(i^{4}\\right)^{4} i+(i^{4})^{4} \\cdot i^{2}-\\left(i^{4}\\right)^{5} \\cdot i^{3}$"
] | |
HW | [
"$\\displaystyle =1-i+(-1)-(-i)$"
] | |
HW | [
"$\\displaystyle =1-i+(-1)-(-i)$"
] | |
HW | [
"$\\displaystyle =0$"
] | |
HW | [
"$\\displaystyle =0$"
] | |
HW | [
"0"
] | |
HW | [
"0"
] | |
HW | [
"$\\displaystyle 16-4 a+40-4 a=0$"
] | |
HW | [
"$\\displaystyle 16-4 a+40-4 a=0$"
] | |
HW | [
"$\\displaystyle 56=8 a, a=7$"
] | |
HW | [
"$\\displaystyle 56=8 a, a=7$"
] | |
HW | [
"$\\displaystyle 49 k-56+10=0$"
] | |
HW | [
"$\\displaystyle 49 k-56+10=0$"
] | |
HW | [
"$\\displaystyle 49 k=46$"
] | |
HW | [
"$\\displaystyle 49 k=46$"
] | |
HW | [
"$\\displaystyle k=\\frac{46}{49}$"
] | |
HW | [
"$\\displaystyle a \\times k=7\\times \\frac{46}{49}=\\frac{46}{7}$"
] | |
HW | [
"$\\displaystyle a \\times k=7\\times \\frac{46}{49}=\\frac{46}{7}$"
] | |
HW | [
"$\\displaystyle \\frac{46}{7}$"
] | |
HW | [
"$\\displaystyle \\frac{46}{7}$"
] | |
HW | [
"$\\displaystyle \\alpha-1+\\frac{1}{\\alpha-1}$"
] |
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