Lemma

Algebra · Chapter 11 of 20

Quadratics

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Endless problems made on demand, each with an exactly computed answer. Every click is a new one.

Solve a quadraticMedium · 83,653 variations

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60
#TitleDifficultySourceTags
49The lowest pointMediumLemmaCompleting the square
1206Quadratic Equation with Binomial RootsMediumHall & KnightQuadratic equationsRoots
1209Quadratic Equation with Parametric Surd RootsMediumHall & KnightQuadratic equationsParameters+1
1212Quadratic from Pure Imaginary RootsMediumHall & KnightQuadratic equationsPure imaginary roots+1
1217Equal roots parameter values 2MediumHall & Knightdiscriminantequal roots
1218Roots equal in magnitude and opposite in signMediumHall & Knightroots of quadraticssum of roots
1220Symmetric function of roots 4MediumHall & Knightroots of quadratic equationVieta's formulas+1
1221Difference of roots squared ratioMediumHall & Knightroots of quadratic equationVieta's formulas
1225Form equation with derived rootsMediumHall & Knightquadratic equationssum and product of roots
1227Condition for multiple rootsMediumHall & Knightroots of a quadraticVieta's formulas
1228Form equation with modified rootsMediumHall & Knightquadratic equationsVieta's formulas+1
1229Equation from sum and difference squaresMediumHall & Knightquadratic equationssum and difference of roots
1230Signs of quadratic rootsMediumHall & Knightquadratic equationsDescartes' rule of signs+1
1231Limits for real rootsMediumHall & Knightquadratic equationsdiscriminant+1
1232Bounds of a rational fractionMediumHall & Knightrational functionsdiscriminant+1
1233Bounds of a symmetric fractionMediumHall & Knightrational functionsdiscriminant+1
1234Excluded values of a fractionMediumHall & Knightrational functionsdiscriminant+1
1235Equation from modified rootsMediumHall & Knightroots of equationssum and product of roots+1
1236Ratio of roots identityMediumHall & KnightQuadratic equationsRoots and coefficients
1237Range of a rational expressionMediumHall & KnightRange of rational functionsDiscriminant method
1238Difference of roots invarianceMediumHall & KnightDiscriminantRoots of quadratic equations
1239Parameter bounds for real valuesMediumHall & KnightRange of rational functionsDiscriminant method
1240Maximum value of a fractionMediumHall & KnightExtremaDiscriminant method
1241Excluded values of a rational expressionMediumHall & KnightRange of rational functionsDiscriminant method
1242Condition for Impossible RootsMediumHall & Knightquadratic equationsdiscriminant+1
1244Resolution into rational factorsMediumHall & Knightquadratic equationsdiscriminant+1
1245Linear factors of quadratic in two variablesMediumHall & Knightquadratic equationsdiscriminant+1
1246Real linear factorsMediumHall & Knightquadratic equationsdiscriminant+1
1247Common root of two quadraticsMediumHall & Knightquadratic equationscommon root
1249Real values in a quadratic relationMediumHall & Knightquadratic equationsdiscriminant+1
1250Bounds for variables in a quadratic relationMediumHall & Knightquadratic equationsdiscriminant+1
1254Radical exponent equationMediumHall & Knightfractional exponentssubstitution
1260Quintic Radical PowersMediumHall & Knightrational exponentsquadratic form
1267Product of binomials equationMediumHall & Knightpolynomial equationsrearrangement of factors
1268Quartic equation by grouping factorsMediumHall & Knightpolynomial equationsrearrangement of factors
1269Solve Quartic EquationMediumHall & Knightquartic equationfactor grouping
1270Solve Polynomial EquationMediumHall & Knightpolynomial equationgrouping factors
1271Radical Equation SubstitutionMediumHall & Knightradical equationsubstitution
1272Radical Equation with Quadratic FormMediumHall & Knightradical equationsubstitution
1273Radical Equation ReductionMediumHall & Knightradical equationsubstitution
1274Radical and Quadratic EquationMediumHall & Knightradical equationsubstitution
1275Rationalizing radical equationMediumHall & Knightradical equationsextraneous roots
1277Difference of radicalsMediumHall & Knightradical equationsfactoring
1278Sum of three square rootsMediumHall & Knightradical equations
1280Homogeneous radical equation with parameterMediumHall & Knightradical equationsparametric equations
1281Radical Equation SubstitutionMediumHall & Knightradicalssubstitution+1
1282Radical Equation SumMediumHall & Knightradicalssubstitution+1
1283Radical Difference EquationMediumHall & Knightradicalsquadratic equations
1287Reciprocal quartic equationMediumHall & Knightquartic equationsubstitution+1
1288Symmetric quartic equationMediumHall & Knightquartic equationsubstitution+1
1289Radical equation with parameterMediumHall & Knightradical equationscomponendo and dividendo
1290Square root ratio equationMediumHall & Knightradical equationscomponendo and dividendo
1292Sum of radicals equationMediumHall & Knightradical equationssimplification
1517Real roots of quadratic equation 1MediumHall & Knightdiscriminantquadratic equations+1
1518Real roots of quadratic equation 2MediumHall & Knightdiscriminantquadratic equations+1
1519Rational roots of quadratic equation 1MediumHall & Knightdiscriminantrational roots
1521Symmetric function of roots part 1MediumHall & Knightroots of a quadraticVieta's formulas+1
1522Symmetric function of roots part 2MediumHall & Knightroots of a quadraticVieta's formulas+1
1523Symmetric functions of roots (Part 1)MediumHall & Knightsymmetric functions of rootsVieta's formulas
1524Symmetric functions of roots (Part 2)MediumHall & Knightsymmetric functions of rootsVieta's formulas

From open sources

465

Complete problems only, with worked solutions first. Show everything

#TitleDifficulty
19283If x > 1, which of the following decreases as x decreasesMedium
19834If -1 < x < 1, which of the following must be…Medium
19889If -0.5 < x < 0, which of the following must be…Medium
20366If the roots of a quadratic equation are 20 and -7, then…Medium
20784If x,p,q, and r are positive integers and (p)×(q)×(r)=x^2, which of the…Medium
20893T is the set of all positive integers x such that x^2…Medium
21015If x^2 - y^2 = 2xy, then (x+y)^2 =Medium
21638For what value of x between − 5 and 5, inclusive, is…Medium
22100If p and q are the roots of the equation x2 -…Medium
22124Find the roots of the quadratic equation: x2 + 2x - 15…Medium
22299Ujakar and Keshab attempted to solve a quadratic equationMedium
22619If x^2 > x^7> x^5, which of the following could be the…Medium
22886If x^2 = y^3 , where x and y are non negative…Medium
22906If x ≠ 0 and x^3 - (2- x^3)/x^3 = y/x^2, then…Medium
22972What is the absolute value of twice the difference of the roots…Medium
23376If x + y = 2 and x^2 - xy - 10…Medium
24876If x is an integer, which of the following is a possible…Medium
25238If a and b are the roots of the equation x2 -…Medium
25596x,y are odd and z is even then ((x^2+y^2)z^2)/8 isMedium
25824x^2 + bx + 72 = 0 has two distinct integer roots…Medium
26018What is the largest integral value of 'k' for which the quadratic…Medium
26044If -3 < x < 0, which of the following must be…Medium
26204If f(x) = (x^2 - 1)/(x), what is f(1/x) in terms of…Medium
26593x^2 + qx + 72 = 0 has two distinct integer roots…Medium
26669Q is the set of all positive integers x such that x^2…Medium
26719If xy = 3 and x^2 + y^2 = 12, then x/y…Medium
26897How many values of c in x^2 - 5x + c, result…Medium
27106The area of one square is x^2 + 10x + 25 and…Medium
27233If f(x) = ax^3 + x^2 + ax – 2x, then f(b)…Medium
27269If x + y = 6 and x – y = 3…Medium
27982If 8 < x < 9, and x^2 = (10 – y)(10…Medium
28033If x^2+(1/x^2)=6, x^4+(1/x^4)=Medium
28169If f(x)=x^3 and g(x)=x^2, and x is a natural number, what CANNOT…Medium
28729If y = x^2 + d x + 9 does not cut…Medium
29366On a two-dimensional coordinate plane, the line z = x^2 - x^3…Medium
29520x^2-y^2= 16 and 2Medium
30019R is the set of all positive integers x such that x^2…Medium
30721If x^2 = 141, which of the following is closest to a…Medium
30740The area of one square is x^2 + 10x + 25 and…Medium
30815If x > 7, which of the following is equal to (x^2…Medium
30917What is the product of all the possible values of x if…Medium
31062As x increases from 165 to 168, which of the following must…Medium
31187If x + (1/x) = 5, what is the value of T=…Medium
31739If x and y are different integers and x^2 = xy, which…Medium
32082If x^2 + 9/x^2 = 22, what is the value of x…Medium
32398If x≠7 and (x^2 - 49)/(2y) = (x - 7)/4, then in…Medium
32574If y = x^2 + ux + v, y is minimum when…Medium
33065For how many values n is 2^x = x^2 Medium
33472If 20x = 49y, which of the following must be trueMedium
33888If x is an integer and (x)(x^2)(x^3) is positive, which of the…Medium
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