shcards","topicId":"0449c786-8f32-4153-b1b7-bda5031cabca","topicSlug":"algebra-and-functions-6","cards":[{"id":"c20ef1e2-9df6-4c18-867c-2dc76565ec85","cardType":"basic","front":"In $a[f(x)]^2+b[f(x)]+c=0$, what should be replaced by a temporary variable?","back":"The complete repeated expression $f(x)$.","points":1,"mcqOptions":null},{"id":"6eb4cab9-a556-4290-bd5b-19174f10861c","cardType":"basic","front":"What substitution makes $x^6-7x^3+10=0$ quadratic?","back":"$u=x^3$, so $x^6=u^2$.","points":1,"mcqOptions":null},{"id":"352b2072-1be3-4bfd-9409-05003731904b","cardType":"basic","front":"What substitution makes $x-5\\sqrt{x}+6=0$ quadratic?","back":"$u=\\sqrt{x}$, so $x=u^2$.","points":1,"mcqOptions":null},{"id":"6c61d69c-74b5-49f5-9de9-084c5f274862","cardType":"cloze","front":"For $au^2+bu+c=0$, the quadratic formula is $\\text{[...]}$.","back":"For $au^2+bu+c=0$, the quadratic formula is $\\boxed{u=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}}$.","points":null,"mcqOptions":null},{"id":"db16ee4e-bf1a-4849-8e43-1c6f3e3bd32c","cardType":"cloze","front":"If $b^2-4ac>0$, the quadratic in $u$ has [ ].","back":"If $b^2-4ac>0$, the quadratic in $u$ has two distinct real values of $u$.","points":null,"mcqOptions":null},{"id":"e7718cd1-3ffe-45df-ae35-bd6f10cab391","cardType":"cloze","front":"If $b^2-4ac=0$, the quadratic in $u$ has [ ].","back":"If $b^2-4ac=0$, the quadratic in $u$ has one repeated real value of $u$.","points":null,"mcqOptions":null},{"id":"3a107707-7765-4bf0-b481-3ad1cecd5aca","cardType":"cloze","front":"If $b^2-4ac<0$, the quadratic in $u$ has [ ].","back":"If $b^2-4ac<0$, the quadratic in $u$ has no real values of $u$.","points":null,"mcqOptions":null},{"id":"7fe23804-0c2f-4205-80b2-871b9f24df27","cardType":"basic","front":"If $u=\\sqrt{x}$ for real $x$, what restriction applies to $u$?","back":"$u\\geq0$.","points":1,"mcqOptions":null},{"id":"145d31bb-afbd-4ca3-9ce9-a898bf8c5cf5","cardType":"basic","front":"If $u=x^2$ for real $x$, what restriction applies to $u$?","back":"$u\\geq0$.","points":1,"mcqOptions":null},{"id":"40ae2496-f0c9-4520-9864-dbf4589030a6","cardType":"basic","front":"What domain restriction applies to $\\frac{1}{x^2}-\\frac{6}{x}+1=0$?","back":"$x\\neq0$ because denominators cannot be zero.","points":1,"mcqOptions":null},{"id":"54aec714-40fa-4d07-a173-e71a25dd4231","cardType":"cloze","front":"If back-substitution gives $x^2=k$ where $k>0$, then $\\text{[...]}$.","back":"If back-substitution gives $x^2=k$ where $k>0$, then $\\boxed{x=\\pm\\sqrt{k}}$.","points":null,"mcqOptions":null},{"id":"692b5c4a-4413-4ffb-acbc-70195039a129","cardType":"basic","front":"Which two conditions identify factors of $u^2+bu+c$?","back":"

The numbers have product $c$.
The numbers have sum $b$.

","points":2,"mcqOptions":null},{"id":"3282d6d6-0444-48ac-8222-cae668d40d1b","cardType":"cloze","front":"Completing the square gives $\\text{[...]}$.","back":"Completing the square gives $\\boxed{u^2+bu=\\left(u+\\frac{b}{2}\\right)^2-\\left(\\frac{b}{2}\\right)^2}$.","points":null,"mcqOptions":null},{"id":"db44defd-61fc-4934-92a6-280bbfaccf00","cardType":"basic","front":"Solve $(x^2-1)^2-5(x^2-1)+4=0$.","back":"$x=\\pm\\sqrt{2},\\ \\pm\\sqrt{5}$.","points":3,"mcqOptions":null},{"id":"8178670f-73ce-4950-921d-8c04b2fa6305","cardType":"basic","front":"Solve $x-5\\sqrt{x}+6=0$.","back":"$x=4$ or $x=9$.","points":2,"mcqOpt

1.2.5 Algebra and Functions Flashcards

1.2.5 Algebra and Functions Flashcards