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.
Solve a real-coefficient quadratic with complex solutions by completing the square
Solve real-coefficient quadratic equations that have complex solutions.
The equation already presents an isolated squared binomial, so no expansion is needed. We’ll take both complex square roots, rewrite the negative radical with i, undo the horizontal shift with …
Preview problemRewrite quadratic-formula solutions with a negative square root in standard complex form
Solve real-coefficient quadratic equations that have complex solutions.
Standardizing quadratic-formula output requires simplifying the negative radical and distributing the denominator across the entire numerator. We’ll convert the radical to an imaginary term, preserve the plus-minus pair, divide both …
Preview problemVerify that a given complex number is a solution of a quadratic equation
Solve real-coefficient quadratic equations that have complex solutions.
Verification means substituting the candidate everywhere and showing that both complex components cancel. We’ll expand the binomial square including its middle term, reduce i squared, distribute the linear coefficient, combine …
Preview problemConnect x-intercept count to real and complex root fields
Solve real-coefficient quadratic equations that have complex solutions.
The graph reveals real-root information through its intersections with the x-axis. We’ll inspect whether the parabola ever reaches zero, translate that intercept count into the discriminant’s sign, distinguish no real …
Preview problemFind a quadratic equation with real coefficients from a pair of complex conjugate roots
Solve real-coefficient quadratic equations that have complex solutions.
Each root supplies a linear factor, and conjugate roots guarantee the imaginary terms disappear. We’ll form both factors, expose their conjugate structure around the common real part, apply the conjugate …
Preview problemClassify the solutions of a quadratic from its discriminant
Solve real-coefficient quadratic equations that have complex solutions.
Root classification depends first on whether the discriminant is positive, zero, or negative. We’ll compare the given value with zero, interpret the quadratic formula’s plus-minus radical, determine whether the branches …
Preview problemChoose the most direct displayed-form method and solve
Solve real-coefficient quadratic equations that have complex solutions.
The displayed structure should guide the most efficient valid method. We’ll notice the absent linear term, isolate the square in one step, take both complex roots, simplify the negative radical, …
Preview problemFactor a sum of squares over the complex numbers using \(a^2+b^2=(a+bi)(a-bi)\)
Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
A sum of real squares becomes a difference of squares once the second square is written with the imaginary unit. We’ll create the negative square, rewrite the expression in difference …
Preview problemVerify a sum-of-squares factorization over the complex numbers
Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
Verification requires expanding the conjugate factors all the way to an ordinary polynomial. We’ll write all four products, cancel the opposite imaginary middle terms, replace i squared with negative one, …
Preview problemFactor a sum of squares of the form \(x^2+k\) over the complex numbers
Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
Over the complex numbers, a sum of squares can be rewritten as a difference involving an imaginary square. We’ll identify the square root of the constant, attach i, apply the …
Preview problemFactor an expression of the form \(A^2+B^2\) over the complex numbers
Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
Treating the shifted expression as one unit keeps the factorization organized. We’ll substitute a temporary variable for the whole binomial, split the resulting sum of squares into conjugates, restore the …
Preview problemSolve \(u^2+b^2=0\) by factoring into complex conjugates
Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
Factoring over the complex numbers turns the sum of squares equation into two conjugate linear equations. We’ll form the factors, apply the zero-product property to both, solve each branch, and …
Preview problemDecide whether a proposed algebraic identity is true for complex numbers
Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
Polynomial identities extend to complex numbers because the same distributive and commutative laws apply. We’ll rewrite the square as a product, distribute all four terms, recognize both cross products, combine …
Preview problemFactor polynomials completely over the complex numbers using the sum-of-squares identity
Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
Complete factorization preserves every existing linear factor while splitting any remaining quadratic over the stated field. We’ll retain the original factor, rewrite the sum-of-squares quadratic as conjugates, restore the full …
Preview problemCompare separate real and complex linear factorizations
Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
Whether a polynomial has linear factors depends on the coefficient field. We’ll use the discriminant to rule out real roots, solve for the complex conjugate roots, build their linear factors, …
Preview problemSimplify a product of complex conjugates
Extend polynomial identities to complex numbers, such as factoring sums of squares over the complex numbers.
A complex number times its conjugate is real because the imaginary cross terms cancel. We’ll identify the shared real and opposite imaginary parts, apply the sum-of-squares identity, evaluate the two …
Preview problemDetermine how many complex roots a quadratic polynomial has, counting multiplicity
Know the Fundamental Theorem of Algebra and verify it for quadratic polynomials.
The Fundamental Theorem of Algebra links polynomial degree to the total number of complex roots, counting multiplicity. We’ll read the degree, apply that count, remember that real roots are included …
Preview problemVerify whether a given real or complex number is a root of a quadratic by substitution
Know the Fundamental Theorem of Algebra and verify it for quadratic polynomials.
A candidate is a root exactly when direct substitution makes the polynomial zero. We’ll replace every occurrence of the variable, preserve the signed coefficient, evaluate powers and products before combining …
Preview problemWrite and verify a quadratic factorization from its roots
Know the Fundamental Theorem of Algebra and verify it for quadratic polynomials.
The root-to-factor rule reverses each root’s sign inside its linear factor. We’ll translate both roots, expand the product, combine the linear terms, and compare every coefficient with the target polynomial …
Preview problemIdentify the repeated root and its multiplicity in a quadratic of the form a(x-r)^2
Know the Fundamental Theorem of Algebra and verify it for quadratic polynomials.
A repeated root and its multiplicity are different pieces of information. We’ll expose the squared linear factor as two copies, solve one copy for the root value, read the repetition …
Preview problemSeparate quadratic root type, distinctness, and multiplicity counts
Know the Fundamental Theorem of Algebra and verify it for quadratic polynomials.
The discriminant controls root type and distinctness, while polynomial degree controls total multiplicity. We’ll classify the positive nonzero discriminant, distinguish the formula’s two branches, identify the roots as distinct, and …
Preview problemFind the other root of a quadratic with real coefficients when one complex root is given
Know the Fundamental Theorem of Algebra and verify it for quadratic polynomials.
A real-coefficient polynomial pairs every nonreal root with its complex conjugate. We’ll confirm the imaginary part is nonzero, keep the real component fixed, reverse only the imaginary sign, and check …
Preview problemWrite a quadratic equation from two given roots
Know the Fundamental Theorem of Algebra and verify it for quadratic polynomials.
Building a monic quadratic from roots starts with one factor x minus r for each root. We’ll form both factors, expand their product, combine the middle terms, attach the equation …
Preview problemDecide whether a claimed root list accounts for all roots of a quadratic polynomial, counting multiplicity
Know the Fundamental Theorem of Algebra and verify it for quadratic polynomials.
A complete root list must account for multiplicity, not only distinct values. We’ll factor the quadratic as a perfect square, identify the single zero of its repeated factor, read the …
Preview problemTranslate quadratic graph behavior into root values and multiplicities
Know the Fundamental Theorem of Algebra and verify it for quadratic polynomials.
Graphical roots are the x-coordinates of x-axis intercepts, and crossing behavior carries multiplicity information. We’ll read both x-values, note that the curve passes through rather than turns back, infer odd …
Preview problemRewrite an expression with exponent \(1/n\) in radical notation
Explain rational exponents as extensions of integer exponent rules and as radical notation.
In a unit-fraction exponent, the denominator becomes the radical’s index. We’ll identify that denominator, translate it to the corresponding root, apply the convention that a square-root index is omitted, and …
Preview problemRewrite a rational exponent in radical notation
Explain rational exponents as extensions of integer exponent rules and as radical notation.
A rational exponent assigns different jobs to its numerator and denominator. We’ll use the denominator as the root index, use the numerator as the power, build the radical without swapping …
Preview problemRewrite a radical as a rational exponent by matching the root index to the denominator
Explain rational exponents as extensions of integer exponent rules and as radical notation.
To convert a radical to one power, the inner exponent becomes the numerator and the root index becomes the denominator. We’ll rewrite the radical as a fractional power, multiply the …
Preview problemEvaluate a rational exponent by using the denominator as the root index
Explain rational exponents as extensions of integer exponent rules and as radical notation.
A rational exponent combines a root and a power, with the denominator naming the root and numerator naming the power. We’ll rewrite the expression accordingly, evaluate the principal root first …
Preview problemEvaluate powers with negative rational exponents by using roots and reciprocals
Explain rational exponents as extensions of integer exponent rules and as radical notation.
The negative sign on an exponent makes the entire positive power reciprocal. We’ll remove that sign by placing the power in the denominator, interpret the remaining fractional exponent as a …
Preview problemDecide whether two rational-exponent expressions are equivalent
Explain rational exponents as extensions of integer exponent rules and as radical notation.
Equivalent fractional exponents represent the same power wherever both expressions are defined. We’ll reduce the exponent fraction, compare the resulting powers directly, connect both to the same radical, and state …
Preview problemDecide whether a negative number raised to a rational exponent is real
Explain rational exponents as extensions of integer exponent rules and as radical notation.
For a negative base, the reduced exponent denominator determines whether the corresponding real root exists. We’ll keep the parentheses around the base, identify the odd root index, rewrite the power …
Preview problemInterpret a rational exponent's rooted quantity and dimension
Explain rational exponents as extensions of integer exponent rules and as radical notation.
The one-half power reverses the square relationship between a square’s side and area. We’ll write the area formula, interpret the exponent as a principal square root, explain why the geometric …
Preview problemReduce a rational exponent, then evaluate it exactly
Explain rational exponents as extensions of integer exponent rules and as radical notation.
Reducing a rational exponent first can reveal a familiar exact root. We’ll divide numerator and denominator by their common factor, preserve the reduced exponential form, translate its denominator into a …
Preview problemSimplify a product of like bases by adding rational exponents
Rewrite expressions with radicals and rational exponents using exponent properties.
Multiplying powers with the same base means adding their exponents, including fractional ones. We’ll keep the base, add the rational exponents with their common denominator, simplify the sum, and carry …
Preview problemSimplify a quotient with rational exponents by subtracting exponents
Rewrite expressions with radicals and rational exponents using exponent properties.
Dividing powers with the same nonzero base subtracts the denominator exponent from the numerator exponent. We’ll keep that order, simplify the fractional difference, use the resulting first power, and retain …
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