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.
Write an equation from a graph description by choosing the correct model form
Create equations in two or more variables and graph them with appropriate labels and scales.
Because the graph is a straight line, slope-intercept form connects its two stated features directly to an equation. We’ll place the rise-per-unit value in the slope position and the vertical-axis …
Preview problemWrite an equation from a table and describe the graph features
Create equations in two or more variables and graph them with appropriate labels and scales.
The change pattern tells us which family fits before we write an equation. We’ll compare consecutive y-values over equal one-unit changes in x, use a constant first difference as the …
Preview problemBuild and graph a constraint equation or inequality
Create equations in two or more variables and graph them with appropriate labels and scales.
The budget constraint starts by adding each item’s unit cost times its count. We’ll translate “at most” with an inclusive inequality, use the equality case to draw a solid boundary …
Preview problemRearrange a formula to solve for a chosen variable
Rearrange formulas to isolate a chosen quantity, including formulas with quadratic terms.
To isolate x, we reverse the operations applied to it in the opposite order. We’ll remove the added constant from both sides first, then divide the entire remaining difference by …
Preview problemRearrange a formula to isolate one variable using inverse operations
Rearrange formulas to isolate a chosen quantity, including formulas with quadratic terms.
The target variable sits inside a sum, and that whole sum is multiplied by two, so we undo operations from the outside inward. We’ll remove the outside factor first, then …
Preview problemSolve a formula for a variable in the denominator
Rearrange formulas to isolate a chosen quantity, including formulas with quadratic terms.
Because the target variable is in a denominator, we first record the nonzero restrictions and clear the fraction by multiplication. That turns the formula into a product equation, which we …
Preview problemRearrange a formula with a squared variable and choose the meaningful square-root branch
Rearrange formulas to isolate a chosen quantity, including formulas with quadratic terms.
Undoing a square algebraically produces both a positive and a negative square-root branch. We’ll write both branches first, then use the fact that s represents a geometric side length to …
Preview problemRearrange a formula with a squared expression to solve for the variable inside the square
Rearrange formulas to isolate a chosen quantity, including formulas with quadratic terms.
The variable is buried inside a shifted square, so we isolate that entire squared expression before taking a root. We’ll remove the outside shift, divide by the nonzero multiplier, retain …
Preview problemSolve a formula when the target variable appears in more than one term
Rearrange formulas to isolate a chosen quantity, including formulas with quadratic terms.
Since the target variable appears in two terms, dividing immediately would miss part of its coefficient. We’ll factor the repeated variable from both terms, treat the remaining sum as one …
Preview problemRearrange a formula to isolate a chosen variable
Rearrange formulas to isolate a chosen quantity, including formulas with quadratic terms.
The target m is multiplied by x before the constant is added, so we reverse those operations in the opposite order. We’ll subtract the intercept term first, then divide the …
Preview problemRearrange a multiplicative formula to isolate a chosen variable, then evaluate it
Rearrange formulas to isolate a chosen quantity, including formulas with quadratic terms.
This asks for a symbolic rearrangement and then a numerical evaluation, so the order matters. We’ll first isolate t by dividing by the nonzero rate, then substitute the given distance …
Preview problemDecide whether two rearranged formulas are equivalent when solving for a variable
Rearrange formulas to isolate a chosen quantity, including formulas with quadratic terms.
To compare the two rearrangements, appearance is less useful than rewriting one into the other. We’ll split the fraction in the first form across every term of its numerator, simplify …
Preview problemState restrictions when rearranging a formula
Rearrange formulas to isolate a chosen quantity, including formulas with quadratic terms.
Restrictions come from both the original formula and the operations used while rearranging it. We’ll protect the original denominator, inspect the new denominator created when isolating t, and require the …
Preview problemComplete the square for a quadratic of the form x^2 + bx + c
Complete the square to transform quadratics and derive the quadratic formula.
Completing the square means building a perfect-square trinomial without changing the expression’s value. We’ll take half the linear coefficient and square it, add and subtract that same amount, factor the …
Preview problemComplete the square for a quadratic when the leading coefficient is not 1
Complete the square to transform quadratics and derive the quadratic formula.
Because the leading coefficient is not one, we first factor it from both variable terms before completing the square inside the grouping. We’ll halve the new linear coefficient, add its …
Preview problemRewrite a quadratic in standard form as vertex form by completing the square
Complete the square to transform quadratics and derive the quadratic formula.
Vertex form comes from turning the quadratic and linear terms into one perfect square. We’ll halve the linear coefficient and square it, add and subtract that value to preserve equivalence, …
Preview problemSolve a monic quadratic equation by completing the square
Complete the square to transform quadratics and derive the quadratic formula.
Completing the square lets us replace the quadratic and linear terms with one squared binomial. We’ll move the constant, add the square of half the linear coefficient to both sides, …
Preview problemFind the term to add to x^2 + bx to complete the square
Complete the square to transform quadratics and derive the quadratic formula.
The missing constant must make the first three terms match a binomial square. We’ll take half of the linear coefficient and square that half, then verify the value by factoring …
Preview problemFind the value to add to \(x^2 + bx\) to complete the square
Complete the square to transform quadratics and derive the quadratic formula.
For a monic expression, the completing value is not the linear coefficient itself or its square. We’ll halve the coefficient first, square the result, and confirm that adding this value …
Preview problemAdd the completing-square term when deriving the quadratic formula
Complete the square to transform quadratics and derive the quadratic formula.
At this point in the quadratic-formula derivation, the left side is monic and its full linear coefficient is the fraction b over a. We’ll halve that entire fraction, square it, …
Preview problemFind the term to add when completing the square in x^2 + (b/a)x
Complete the square to transform quadratics and derive the quadratic formula.
The completing term comes from the full fractional coefficient of x. We’ll divide that coefficient by two, square both its numerator and denominator, simplify the resulting fraction, and verify it …
Preview problemIdentify the maximum or minimum of a quadratic written in vertex form
Complete the square to transform quadratics and derive the quadratic formula.
Vertex form separates the parabola’s turning point from the coefficient that controls its opening. We’ll read h by reversing the sign inside the parentheses, use k as the vertex’s output, …
Preview problemComplete the square exactly for a monic quadratic with a fractional x-coefficient
Complete the square to transform quadratics and derive the quadratic formula.
Fractional coefficients use the same completing-square idea, but every value must stay exact. We’ll halve the fractional x coefficient, square that fraction, add and subtract the result, factor the perfect …
Preview problemTransform a monic quadratic equation into a perfect-square equation by completing the square
Complete the square to transform quadratics and derive the quadratic formula.
The goal is to turn the quadratic side into one perfect-square binomial while keeping the equation equivalent. We’ll find the square of half the linear coefficient, add that same amount …
Preview problemInterpret a completed-square quadratic model in context
Complete the square to transform quadratics and derive the quadratic formula.
Vertex form gives the candidate turning point directly, but the context determines whether that point occurs during the modeled flight. We’ll read time and height from the vertex, compare the …
Preview problemSolve a monic quadratic equation by factoring the trinomial and using the zero-product property
Solve quadratics by inspection, square roots, completing the square, formula, and factoring; express complex solutions.
A monic trinomial factors by finding two numbers whose product is the constant and whose sum is the linear coefficient. We’ll use that pair to write both binomial factors, verify …
Preview problemSolve a non-monic quadratic equation by factoring a trinomial and using the zero-product property
Solve quadratics by inspection, square roots, completing the square, formula, and factoring; express complex solutions.
Because the leading coefficient is not one, we’ll factor by splitting the middle term. We’ll find two numbers whose product is the leading coefficient times the constant and whose sum …
Preview problemSolve a difference-of-squares equation by factoring and using the zero-product property
Solve quadratics by inspection, square roots, completing the square, formula, and factoring; express complex solutions.
This quadratic has the special form one square minus another square. We’ll rewrite the constant as a square, factor the difference into conjugate binomials, and use the zero-product property on …
Preview problemSolve a quadratic equation by taking square roots
Solve quadratics by inspection, square roots, completing the square, formula, and factoring; express complex solutions.
The squared variable is already isolated, so the square-root property is the most direct method. We’ll take both the positive and negative square-root branches, simplify the root, and verify each …
Preview problemSolve a quadratic equation by completing the square
Solve quadratics by inspection, square roots, completing the square, formula, and factoring; express complex solutions.
Completing the square will turn the two variable terms into one squared binomial. We’ll move the constant, add the square of half the linear coefficient to both sides, factor the …
Preview problemSolve a quadratic equation using the quadratic formula
Solve quadratics by inspection, square roots, completing the square, formula, and factoring; express complex solutions.
The quadratic formula works from the signed coefficients in standard form, so sign accuracy matters from the start. We’ll identify a, b, and c, substitute them into the full formula, …
Preview problemUse the discriminant to classify the solutions of a quadratic equation
Solve quadratics by inspection, square roots, completing the square, formula, and factoring; express complex solutions.
The discriminant classifies the roots without requiring us to solve the entire quadratic. We’ll identify the signed coefficients, compute b squared minus four a c, and use whether that value …
Preview problemSolve a quadratic equation with complex solutions by completing the square and writing the answers using i
Solve quadratics by inspection, square roots, completing the square, formula, and factoring; express complex solutions.
After isolating the square, the negative value on the other side tells us the solutions are complex rather than real. We’ll take both square-root branches, separate the negative radicand into …
Preview problemChoose the most efficient method for solving a quadratic equation
Solve quadratics by inspection, square roots, completing the square, formula, and factoring; express complex solutions.
Method choice should use the equation’s current structure before doing any unnecessary algebra. We’ll scan the decision rule in order, recognize that this equation is already a product equal to …
Preview problemSolve a quadratic in factored form using the zero-product property
Solve quadratics by inspection, square roots, completing the square, formula, and factoring; express complex solutions.
The equation is already a product equal to zero, so we can use the zero-product property without expanding. We’ll set each linear factor equal to zero, solve both equations, and …
Preview problemInterpret zeros of a quadratic model in context
Solve quadratics by inspection, square roots, completing the square, formula, and factoring; express complex solutions.
A zero of this signed-elevation model is a position where the output equals the reference level. We’ll set the model equal to zero, keep both square-root branches, check each candidate …
Preview problem