Math II
Connect real and complex numbers, polynomials, quadratics, proof, circles, trigonometry, probability, and modeling.
- Problem types
- 786
- Practice variants
- 3,144
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Each problem type has four distinct practice variants. Open a preview to move among all four.
Factor by grouping
Use expression structure to identify useful rewrites, such as difference-of-squares factoring.
Factoring by grouping succeeds when the term pairs can expose the same binomial. We’ll group the polynomial in pairs, extract each group’s greatest common factor, factor the shared binomial from …
Preview problemFactor completely using a GCF and another pattern
Use expression structure to identify useful rewrites, such as difference-of-squares factoring.
Complete factoring can require more than one pattern, so the greatest common factor comes first. We’ll extract it exactly once, inspect the remaining binomial for a difference of squares, split …
Preview problemRewrite a quadratic to reveal its zeros
Use expression structure to identify useful rewrites, such as difference-of-squares factoring.
Factored form exposes zeros only after the factorization has been justified. We’ll find the integer pair that matches both the constant product and signed linear sum, expand to verify equivalence, …
Preview problemRewrite a quadratic to reveal its vertex
Use expression structure to identify useful rewrites, such as difference-of-squares factoring.
Completing the square rewrites the quadratic without changing its values. We’ll halve the linear coefficient, add and subtract its square as a balanced compensation, combine the perfect-square trinomial and remaining …
Preview problemRewrite an exponential expression to reveal structure
Use expression structure to identify useful rewrites, such as difference-of-squares factoring.
A sum in an exponent becomes a product of same-base powers. We’ll separate the variable and constant exponent parts, evaluate the constant power, absorb it into the outside coefficient, and …
Preview problemChoose the most useful equivalent form
Use expression structure to identify useful rewrites, such as difference-of-squares factoring.
Equivalent forms emphasize different features, so the requested feature should drive the rewrite. We’ll compare what standard, vertex, and factored forms display directly, identify the representation that exposes zeros with …
Preview problemFactor using a repeated subexpression
Use expression structure to identify useful rewrites, such as difference-of-squares factoring.
Treating the repeated binomial as one temporary quantity exposes a difference-of-squares pattern. We’ll factor in that temporary variable, substitute the complete original binomial back into both conjugate factors, simplify each …
Preview problemEvaluate efficiently using expression structure
Use expression structure to identify useful rewrites, such as difference-of-squares factoring.
The expression’s difference-of-squares structure replaces two large square calculations with a small difference and sum. We’ll apply the conjugate identity to the two square bases, evaluate those simple factors, multiply …
Preview problemFactor a monic quadratic to reveal zeros
Factor quadratics to reveal zeros of the function they define.
For a monic quadratic, the binomial constants must match both the constant product and the signed linear sum. We’ll use those two conditions to factor, solve every zero-product branch, and …
Preview problemFactor a nonmonic quadratic to reveal zeros
Factor quadratics to reveal zeros of the function they define.
For a nonmonic quadratic, the ac method turns the middle term into two terms that support grouping. We’ll find a pair with product a c and sum b, split the …
Preview problemFactor a difference of squares to reveal zeros
Factor quadratics to reveal zeros of the function they define.
The subtraction of two perfect squares signals conjugate factors with opposite signs. We’ll identify both square bases, apply the difference-of-squares identity, solve both zero-product branches, and check both candidates in …
Preview problemFactor a perfect-square trinomial to reveal its repeated zero
Factor quadratics to reveal zeros of the function they define.
A perfect-square trinomial represents two identical linear factors, not two distinct zeros. We’ll verify the signed middle term from the endpoint square roots, rewrite the trinomial as one squared binomial, …
Preview problemFactor out a GCF to reveal quadratic zeros
Factor quadratics to reveal zeros of the function they define.
A variable greatest common factor is itself a zero-producing factor, so it must not be divided away. We’ll extract the greatest common monomial, apply the zero-product property to every remaining …
Preview problemIdentify x-intercepts from factored form
Factor quadratics to reveal zeros of the function they define.
An x-intercept is a zero written as a point, so the output must first be set equal to zero. We’ll solve each visible factor, attach a y-coordinate of zero, count …
Preview problemWrite a quadratic from zeros and scale factor
Factor quadratics to reveal zeros of the function they define.
A zero r becomes the factor x minus r, while the scale factor multiplies the entire factor product. We’ll convert each supplied zero with the correct sign, apply the scale, …
Preview problemFactor a quadratic and state its zeros
Factor quadratics to reveal zeros of the function they define.
For a monic trinomial, a valid factor pair must reproduce both the constant product and the signed middle-term sum. We’ll verify those conditions, write the product equal to zero, solve …
Preview problemInterpret zeros of a factored quadratic model
Factor quadratics to reveal zeros of the function they define.
Zeros of the height model are candidate times when the ball is at ground level. We’ll solve the time factors, ignore the nonzero scale as a source of roots, check …
Preview problemDetermine factorability over the integers
Factor quadratics to reveal zeros of the function they define.
Integer factorability requires one signed integer pair to satisfy both the product and sum conditions. We’ll list the relevant factor pairs, test their sums against the middle coefficient, expand any …
Preview problemDetermine a quadratic's sign from factored form
Factor quadratics to reveal zeros of the function they define.
The factor zeros partition the number line into intervals where each factor keeps a constant sign. We’ll order the zeros, determine both factor signs on one point in each interval, …
Preview problemInterpret factors in an area expression
Factor quadratics to reveal zeros of the function they define.
The factors are rectangle dimensions, so each carries length units and both must be positive at the same time. We’ll compare the side expressions, intersect their positivity conditions to obtain …
Preview problemFind a quadratic coefficient from its zeros
Factor quadratics to reveal zeros of the function they define.
The given zeros determine a monic quadratic’s two factors. We’ll convert each negative zero into its x minus r factor, expand the product, match the linear coefficient with the unknown …
Preview problemComplete the square for a monic quadratic
Complete the square to reveal maximum or minimum values of quadratic functions.
Completing the square uses the square of half the signed linear coefficient, with an equal compensation to preserve the expression. We’ll build the perfect-square trinomial, combine the outside constants, read …
Preview problemComplete the square for a nonmonic quadratic
Complete the square to reveal maximum or minimum values of quadratic functions.
For a nonmonic quadratic, completing the square starts by factoring the leading coefficient from only the variable terms. We’ll complete the square inside that group, remember that the outside coefficient …
Preview problemIdentify a vertex from completed-square form
Complete the square to reveal maximum or minimum values of quadratic functions.
Vertex form exposes the horizontal shift, vertical shift, and scale in separate parameters. We’ll match the expression with a times the quantity x minus h squared plus k, use h …
Preview problemClassify a quadratic extremum
Complete the square to reveal maximum or minimum values of quadratic functions.
Minimum versus maximum is determined by the leading coefficient’s sign, not by whether the vertex output is positive or negative. We’ll use that sign for the opening, read the vertex, …
Preview problemAnalyze extrema from vertex form
Complete the square to reveal maximum or minimum values of quadratic functions.
The nonnegative-square fact gives an algebraic bound without relying only on the graph. We’ll identify where the square reaches equality, shift its lower bound by the outside constant, and use …
Preview problemOptimize a quadratic model by completing the square
Complete the square to reveal maximum or minimum values of quadratic functions.
Completing the square turns the revenue model’s optimum into the vertex of a downward-opening parabola. We’ll factor the negative leading coefficient, complete and compensate the square inside, distribute the sign …
Preview problemFind the axis of symmetry of a quadratic by writing it in vertex form
Complete the square to reveal maximum or minimum values of quadratic functions.
The function is already in vertex form, so the symmetry axis comes from the center of the squared displacement. We’ll read h with the sign convention x minus h, locate …
Preview problemExtract graph features from completed-square form
Complete the square to reveal maximum or minimum values of quadratic functions.
Vertex form supplies the center and scale, while symmetry lets one offset predict its matching point. We’ll read the vertex, axis, opening, width, and extremum, then evaluate inputs the same …
Preview problemDetermine range from vertex form
Complete the square to reveal maximum or minimum values of quadratic functions.
A real square’s nonnegative bound becomes a bound on the function after the vertical shift is applied. We’ll locate the input where equality occurs, confirm that the boundary output is …
Preview problemIdentify an extremum from completed-square form
Complete the square to reveal maximum or minimum values of quadratic functions.
The squared term can contribute no less than zero, and it reaches zero only at the center input. We’ll shift that bound by the outside constant, use the equality case …
Preview problemRewrite an exponential model when the input is measured in a different unit
Use exponent properties to transform exponential expressions and interpret growth/decay rates.
Changing the input unit changes how many original growth intervals one new interval contains. We’ll express months as a multiple of years, substitute that count into the exponent, regroup the …
Preview problemRewrite an exponential model when the input unit changes
Use exponent properties to transform exponential expressions and interpret growth/decay rates.
A month is one twelfth of a year, so changing the input unit requires replacing the yearly interval count by a fractional year count. We’ll substitute that conversion, regroup the …
Preview problemRewrite an exponential expression with a new base using the power-of-a-power rule
Use exponent properties to transform exponential expressions and interpret growth/decay rates.
To change the exponential base, first express the old base as an exact power of the requested one. We’ll substitute that power, multiply the nested exponents with the power-of-a-power rule, …
Preview problemInterpret percent growth from an exponential model
Use exponent properties to transform exponential expressions and interpret growth/decay rates.
The exponential base gives the full amount retained after each interval, not just the change. We’ll separate the base into one plus a decimal rate, convert that added fraction to …
Preview problemInterpret percent decay from an exponential model
Use exponent properties to transform exponential expressions and interpret growth/decay rates.
For decay, the base represents the fraction retained each interval, while the loss is its complement to one. We’ll classify the base between zero and one, convert the retained fraction …
Preview problem