Algebra · Chapter 11 of 20

Quadratics

89 curated · 949 from open sources · 104.5 thousand generated

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104.5 thousand

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

Solve a quadraticMedium · 83,653 variations Quadratic with fractional rootsDeep · 20,808 variations

Curated

#	Title	Difficulty	Source	Tags
105	Factor it	Easy	Lemma	FactoringZero product
747	Factor and solve	Easy	Lemma	Quadratics
748	The vertex	Easy	Lemma	Quadratics
749	The discriminant	Easy	Lemma	Quadratics
750	Product of roots	Easy	Lemma	Quadratics
1204	Quadratic Equation from Rational Roots	Easy	Hall & Knight	Quadratic equationsRoots
1205	Quadratic Equation from Algebraic Roots	Easy	Hall & Knight	Quadratic equationsRoots
1207	Quadratic Equation with Surd Roots	Easy	Hall & Knight	Quadratic equationsSurds+1
1208	Quadratic Equation with Offset Surds	Easy	Hall & Knight	Quadratic equationsSurds+1
1210	Quadratic from Complex Roots	Easy	Hall & Knight	Quadratic equationsComplex roots+1
1211	Quadratic from General Complex Roots	Easy	Hall & Knight	Quadratic equationsComplex roots+1
1216	Equal roots parameter values 1	Easy	Hall & Knight	discriminantequal roots
1219	Sum of inverse squares of roots	Easy	Hall & Knight	roots of quadraticsVieta's formulas
1226	Real roots of quadratic equation	Easy	Hall & Knight	quadratic equationsdiscriminant+1
1251	Equation reducible to quadratic form	Easy	Hall & Knight	substitutionquadratic equations
1252	Biquadratic reciprocal equation	Easy	Hall & Knight	substitutionnegative exponents
1253	Equation with fractional exponents	Easy	Hall & Knight	fractional exponentsquadratic equations
1257	Radical Equation Substitution	Easy	Hall & Knight	radical equationssubstitution
1258	Reciprocal Radical Equation	Easy	Hall & Knight	radical equationssubstitution
1259	Fractional Exponents Equation	Easy	Hall & Knight	fractional exponentsquadratic form
1864	Minimum Value of Quadratic	Easy	Hall & Knight	Quadratic extremaCompleting the square
1865	Maximum Value of Quadratic	Easy	Hall & Knight	Quadratic extremaCompleting the square
49	The lowest point	Medium	Lemma	Completing the square
1206	Quadratic Equation with Binomial Roots	Medium	Hall & Knight	Quadratic equationsRoots
1209	Quadratic Equation with Parametric Surd Roots	Medium	Hall & Knight	Quadratic equationsParameters+1
1212	Quadratic from Pure Imaginary Roots	Medium	Hall & Knight	Quadratic equationsPure imaginary roots+1
1217	Equal roots parameter values 2	Medium	Hall & Knight	discriminantequal roots
1218	Roots equal in magnitude and opposite in sign	Medium	Hall & Knight	roots of quadraticssum of roots
1220	Symmetric function of roots 4	Medium	Hall & Knight	roots of quadratic equationVieta's formulas+1
1221	Difference of roots squared ratio	Medium	Hall & Knight	roots of quadratic equationVieta's formulas
1225	Form equation with derived roots	Medium	Hall & Knight	quadratic equationssum and product of roots
1227	Condition for multiple roots	Medium	Hall & Knight	roots of a quadraticVieta's formulas
1228	Form equation with modified roots	Medium	Hall & Knight	quadratic equationsVieta's formulas+1
1229	Equation from sum and difference squares	Medium	Hall & Knight	quadratic equationssum and difference of roots
1230	Signs of quadratic roots	Medium	Hall & Knight	quadratic equationsDescartes' rule of signs+1
1231	Limits for real roots	Medium	Hall & Knight	quadratic equationsdiscriminant+1
1232	Bounds of a rational fraction	Medium	Hall & Knight	rational functionsdiscriminant+1
1233	Bounds of a symmetric fraction	Medium	Hall & Knight	rational functionsdiscriminant+1
1234	Excluded values of a fraction	Medium	Hall & Knight	rational functionsdiscriminant+1
1235	Equation from modified roots	Medium	Hall & Knight	roots of equationssum and product of roots+1
1236	Ratio of roots identity	Medium	Hall & Knight	Quadratic equationsRoots and coefficients
1237	Range of a rational expression	Medium	Hall & Knight	Range of rational functionsDiscriminant method
1238	Difference of roots invariance	Medium	Hall & Knight	DiscriminantRoots of quadratic equations
1239	Parameter bounds for real values	Medium	Hall & Knight	Range of rational functionsDiscriminant method
1240	Maximum value of a fraction	Medium	Hall & Knight	ExtremaDiscriminant method
1241	Excluded values of a rational expression	Medium	Hall & Knight	Range of rational functionsDiscriminant method
1242	Condition for Impossible Roots	Medium	Hall & Knight	quadratic equationsdiscriminant+1
1244	Resolution into rational factors	Medium	Hall & Knight	quadratic equationsdiscriminant+1
1245	Linear factors of quadratic in two variables	Medium	Hall & Knight	quadratic equationsdiscriminant+1
1246	Real linear factors	Medium	Hall & Knight	quadratic equationsdiscriminant+1
1247	Common root of two quadratics	Medium	Hall & Knight	quadratic equationscommon root
1249	Real values in a quadratic relation	Medium	Hall & Knight	quadratic equationsdiscriminant+1
1250	Bounds for variables in a quadratic relation	Medium	Hall & Knight	quadratic equationsdiscriminant+1
1254	Radical exponent equation	Medium	Hall & Knight	fractional exponentssubstitution
1260	Quintic Radical Powers	Medium	Hall & Knight	rational exponentsquadratic form
1267	Product of binomials equation	Medium	Hall & Knight	polynomial equationsrearrangement of factors
1268	Quartic equation by grouping factors	Medium	Hall & Knight	polynomial equationsrearrangement of factors
1269	Solve Quartic Equation	Medium	Hall & Knight	quartic equationfactor grouping
1270	Solve Polynomial Equation	Medium	Hall & Knight	polynomial equationgrouping factors
1271	Radical Equation Substitution	Medium	Hall & Knight	radical equationsubstitution
1272	Radical Equation with Quadratic Form	Medium	Hall & Knight	radical equationsubstitution
1273	Radical Equation Reduction	Medium	Hall & Knight	radical equationsubstitution
1274	Radical and Quadratic Equation	Medium	Hall & Knight	radical equationsubstitution
1275	Rationalizing radical equation	Medium	Hall & Knight	radical equationsextraneous roots
1277	Difference of radicals	Medium	Hall & Knight	radical equationsfactoring
1278	Sum of three square roots	Medium	Hall & Knight	radical equations
1280	Homogeneous radical equation with parameter	Medium	Hall & Knight	radical equationsparametric equations
1281	Radical Equation Substitution	Medium	Hall & Knight	radicalssubstitution+1
1282	Radical Equation Sum	Medium	Hall & Knight	radicalssubstitution+1
1283	Radical Difference Equation	Medium	Hall & Knight	radicalsquadratic equations
1287	Reciprocal quartic equation	Medium	Hall & Knight	quartic equationsubstitution+1
1288	Symmetric quartic equation	Medium	Hall & Knight	quartic equationsubstitution+1
1289	Radical equation with parameter	Medium	Hall & Knight	radical equationscomponendo and dividendo
1290	Square root ratio equation	Medium	Hall & Knight	radical equationscomponendo and dividendo
1292	Sum of radicals equation	Medium	Hall & Knight	radical equationssimplification
1517	Real roots of quadratic equation 1	Medium	Hall & Knight	discriminantquadratic equations+1
1518	Real roots of quadratic equation 2	Medium	Hall & Knight	discriminantquadratic equations+1
1519	Rational roots of quadratic equation 1	Medium	Hall & Knight	discriminantrational roots
1521	Symmetric function of roots part 1	Medium	Hall & Knight	roots of a quadraticVieta's formulas+1
1522	Symmetric function of roots part 2	Medium	Hall & Knight	roots of a quadraticVieta's formulas+1
1523	Symmetric functions of roots (Part 1)	Medium	Hall & Knight	symmetric functions of rootsVieta's formulas
1524	Symmetric functions of roots (Part 2)	Medium	Hall & Knight	symmetric functions of rootsVieta's formulas
1243	Range of a Rational Expression	Deep	Hall & Knight	quadratic equationsreal roots condition+1
1248	Parameter for linear factors	Deep	Hall & Knight	quadratic equationsdeterminant condition+1
1276	Algebraic equation with radicals	Deep	Hall & Knight	radical equationsalgebraic manipulation
1279	Linear combination of radicals	Deep	Hall & Knight	radical equationsfactoring quadratics
1284	Radicals with Linear Term	Deep	Hall & Knight	radicalsfactoring+1
1291	Equation with radical differences	Deep	Hall & Knight	rationalizationradical equations
1520	Rational roots of quadratic equation 2	Deep	Hall & Knight	discriminantrational roots

From open sources

949

Complete problems only, with worked solutions first. Show everything

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