Algebra · Chapter 11 of 20

Quadratics

60 curated · 1,022 from open sources · 83,653 generated

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Solve a quadraticMedium · 83,653 variations

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#	Title	Difficulty	Source	Tags
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

From open sources

1,022

Showing everything, including problems with no answer to check, a missing figure, or a near-copy elsewhere. Show only complete problems

#	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