← PSAT/NMSQT®

Math: Advanced Math

About 32.5 percent of PSAT math: quadratics, exponentials and polynomials, plus the one form of each expression that makes it give up its secret.

38 topics · 160 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Seeing Structure in an Expression

  1. How it worksHow does treating a repeated group as one block turn a messy expression into a simple one?
  2. Worked exampleShow me how 3(x + 4)2 - 12 reveals its smallest value once it is read as a block.
  3. Worked exampleFind the value of (a + b)2 when a + b equals 7, without ever knowing a or b.
  4. Common mistakeWhy isn't an expanded expression always simpler than a factored one?
Your turnmedium1 of 3
A problem asks for the value of (7m + 7n)2 - 9(7m + 7n), and tells you that 7m + 7n = 6. Find that value.

Adding and Subtracting Polynomials

  1. How it worksHow does subtracting a polynomial flip the sign of every single term inside the brackets?
  2. Worked exampleFind (4x2 - 3x + 7) minus (x2 + 5x - 2) as one collected polynomial.
  3. Common mistakeWhy isn't 3x2 + 4x equal to 7x3?
Your turneasy1 of 3
Find (10x2 - 7x + 9) - (2x2 + 2x - 3) as one collected polynomial in x.

Multiplying Polynomials

  1. How it worksHow does every term in one bracket have to meet every term in the other?
  2. Common mistakeWhy isn't (x + 6)2 the same as x2 + 36?
  3. Worked exampleFind the product (2x - 5)(3x + 4) and collect the like terms.
  4. Worked exampleFind (x + 6)2 expanded, and show exactly where the middle term comes from.
Your turneasy1 of 3
Expand (5x - 7)(7x + 4) and collect the like terms, in terms of x.

Factoring Out a Common Factor

  1. How it worksHow do you spot the largest factor that every term in an expression shares?
  2. Worked exampleFind the greatest common factor of 12x3 - 18x2 and pull it out completely.
  3. Worked exampleShow me 5x(x - 2) + 3(x - 2) factored by treating x - 2 as the shared piece.
  4. Going furtherWhat happens to the brackets when the common factor you pull out is negative?
Your turneasy1 of 3
Factor 7x(x - 7) + 9(x - 7) completely, in terms of x.

The Difference of Two Squares

  1. How it worksHow does the middle term cancel itself out when (a + b)(a - b) is multiplied?
  2. Worked exampleShow me 49x2 - 25 factored into two binomials.
  3. Common mistakeWhy doesn't x2 + 16 factor the way x2 - 16 does?
  4. Worked exampleFind 103 times 97 in one step by reading it as a difference of squares.
Your turneasy1 of 3
Factor 72x2 - 82 into two binomials, in terms of x.

Factoring a Trinomial into Two Binomials

  1. How it worksHow do the two numbers in the brackets control the middle term and the constant at once?
  2. Worked exampleFind the two binomials whose product is x2 - 7x + 12.
  3. Worked exampleFind the factored form of 2x2 + 11x + 12, where the leading coefficient is not 1.
  4. Common mistakeWhy doesn't every trinomial break into two neat binomials with whole numbers in them?
  5. Going furtherWhat changes about the sign pattern when the constant term is negative?
Your turneasy1 of 3
Multiply out (x - 7)(x - 8) and write the result as a trinomial in standard form.

Solving a Quadratic Given in Factored Form

  1. How it worksHow does each bracket of a factored quadratic hand you one crossing point directly?
  2. Common mistakeWhy doesn't (x - 3)(x + 2) = 6 mean that x - 3 is 6 or x + 2 is 6?
  3. Worked exampleFind both solutions of (x - 3)(2x + 7) = 0 and mark them on the parabola.
  4. Worked exampleFind the x-intercepts of y = (x + 5)(x - 1) and the axis of symmetry between them.
Your turneasy1 of 3
Find the larger of the two solutions of (x - 9)(x + 6) = 0.

Solving a Quadratic in Standard Form

  1. How it worksHow does factoring carry a quadratic from standard form to the form that answers the question?
  2. How it worksHow does the zero-product rule stop working the moment one side is something other than zero?
  3. Worked exampleSolve x2 + 2x - 15 = 0 by factoring it first.
  4. Worked exampleSolve 3x2 - 12x = 0 by pulling out the common factor instead of reaching for a formula.
  5. Common mistakeWhy doesn't dividing x2 = 5x through by x give you the complete solution set?
Your turneasy1 of 3
Solve 7x2 - 14x = 0 and give the solution that is not zero.

Solving a Quadratic Given in Vertex Form

  1. How it worksHow does undoing a square produce two answers where every other step produces one?
  2. Worked exampleSolve 2(x - 4)2 - 18 = 0 by isolating the square and taking the root.
  3. Worked exampleFind the vertex and both roots of y = -(x + 1)2 + 9.
  4. Common mistakeWhy isn't the h in (x - h)2 + k the sign it appears to be inside the bracket?
Your turnmedium1 of 3
Solve 6(x - 8)2 - 56 = 0 and give the larger of the two solutions.

What Completing the Square Actually Is

  1. How it worksHow do an x by x square and two thin rectangles leave a gap of exactly (b/2)2?
  2. How it worksHow does adding and subtracting the same number change an expression's form but not its value?
  3. Worked exampleShow me the square, the two rectangles and the missing corner that build x2 + 6x.
  4. Common mistakeWhy doesn't adding (b/2)2 to one side of an equation alone keep it true?
Your turneasy1 of 3
Find the number that must be added to x2 + 17x to turn it into a perfect square trinomial.

Completing the Square to Solve a Quadratic

  1. How it worksHow does turning a quadratic into a perfect square let you finish in one root step?
  2. Worked exampleSolve x2 + 8x - 9 = 0 by completing the square.
  3. Worked exampleSolve 2x2 - 12x + 10 = 0 by dividing by 2 before completing the square.
  4. Common mistakeWhy isn't completing the square blocked when the middle coefficient is odd?
  5. Going furtherWhat does completing the square hand you when a quadratic has no real roots?
Your turnmedium1 of 3
Solve x2 + 12x - 21 = 0 by completing the square, and give the larger solution.

The Quadratic Formula

  1. How it worksHow does completing the square on ax2 + bx + c produce the quadratic formula itself?
  2. How it worksHow does the plus-or-minus sign in the formula deliver both roots in a single pass?
  3. Worked exampleSolve 3x2 - 5x - 2 = 0 with the quadratic formula, showing every substitution.
  4. Worked exampleSolve x2 + 4x + 1 = 0 and leave both answers in exact radical form.
  5. Common mistakeWhy doesn't the formula work until the equation has been set equal to zero?
Your turnmedium1 of 3
Use the quadratic formula to find the larger solution of 5x2 - 11x - 11 = 0.

Choosing the Fastest Method

  1. How it worksHow do you glance at a quadratic and pick factoring, a square root, or the formula?
  2. Worked exampleSolve 4x2 = 49 in a single move and show why factoring would be slower.
  3. Worked exampleCompare the factoring route against the formula route for x2 - 10x + 21 = 0.
  4. Going furtherWhat changes about your choice of method when the coefficients are decimals?
Your turneasy1 of 3
Solve 7x2 = 137 in one move and give the positive solution.

The Discriminant and the Number of Real Solutions

  1. How it worksHow does b2 - 4ac decide between two crossings, one, or none at all?
  2. Worked exampleFind how many real solutions 2x2 - 4x + 5 = 0 has using the discriminant.
  3. Worked exampleFind the c that gives x2 - 6x + c = 0 exactly one real solution.
  4. Common mistakeWhy doesn't having no real solution mean the parabola vanished from the plane?
  5. Going furtherWhat happens to the two roots as the discriminant shrinks towards zero?
Your turnmedium1 of 3
Find the value of c for which x2 - 17x + c = 0 has exactly one real solution.

Linear Absolute Value Equations

  1. How it worksHow does splitting an absolute value into a positive case and a negative case find both answers?
  2. Worked exampleSolve the absolute value of (2x - 7) equals 9 by splitting it into two equations.
  3. Worked exampleFind every solution of the absolute value of (x + 3) equals -4, and explain the result.
  4. Common mistakeWhy isn't the absolute value of x equal to x for every value of x?
Your turneasy1 of 3
Solve the absolute value of (5x - 14) equals 23, and give the larger solution.

Simple Rational Equations

  1. How it worksHow does multiplying through by the denominator turn a rational equation into a linear one?
  2. Worked exampleSolve 12/(x - 2) = 4 and check the answer against the forbidden value.
  3. Common mistakeWhy doesn't cancelling an x from the top and bottom of a fraction always leave things unchanged?
  4. Worked exampleSolve 3/x + 1 = 5/x by clearing both denominators at once.
Your turneasy1 of 3
Solve 44/(x - 8) = 9.

Radical Equations

  1. How it worksHow does isolating the radical before squaring stop the algebra from exploding?
  2. Common mistakeWhy isn't squaring both sides a reversible move like adding or dividing?
  3. Worked exampleSolve the square root of (2x + 3) equals 5, then verify the answer.
  4. Worked exampleSolve the square root of (x + 6) equals x, and test both candidates.
Your turneasy1 of 3
Solve the square root of (5x + 8) equals 11.

Extraneous Solutions

  1. How it worksHow does squaring erase a minus sign and let a bad candidate slip through?
  2. How it worksHow does clearing a denominator quietly multiply both sides by something that might be zero?
  3. Worked exampleShow me why x = 3 must be rejected in x/(x - 3) = 3/(x - 3) + 2.
  4. Worked exampleFind the real solutions of the square root of (3x + 4) equals x, testing each candidate.
  5. Common mistakeWhy isn't checking your answer optional whenever a radical or a denominator shows up?
Your turnmedium1 of 3
Solve the square root of (22 - x) equals x, and give the solution that survives the check.

Systems of a Linear and a Nonlinear Equation

  1. How it worksHow does substituting the line into the curve collapse a whole system into one quadratic?
  2. Worked exampleSolve y = x2 - 4 together with y = 2x - 1.
  3. Worked exampleFind where y = 3x + 2 meets y = x2 + 2x + 2.
  4. Going furtherWhat has to be true for a line to touch a parabola at exactly one point?
Your turnmedium1 of 3
The graphs of y = x2 - 24 and y = 7x - 8 meet twice. Find the larger of the two x-coordinates.

Relating a System's Solutions to Its Graphs

  1. How it worksHow does the discriminant of the combined equation predict the crossings before anything is drawn?
  2. Worked exampleFind how many times y = x2 + 1 and y = x - 2 meet, using algebra alone.
  3. Common mistakeWhy isn't a system of a line and a curve guaranteed to have any solution at all?
  4. Ties togetherHow does the empty case for a line and a parabola echo two parallel lines never meeting?
Your turneasy1 of 3
Combine y = x2 + 6 with y = 9x - 8 into one quadratic, and find that quadratic's discriminant.

Solving a Formula for the Variable of Interest

  1. How it worksHow do you peel operations off the variable you want while the rest ride along?
  2. Worked exampleSolve A = P(1 + rt) for the rate r.
  3. Worked exampleSolve V = (1/3)Bh for h, and then for B.
Your turneasy1 of 3
Solve V = (1/5)Bh for h, writing h as an expression in V and B.

Creating and Using a Quadratic Function in Context

  1. How it worksHow does the vertex of a height function mark the highest point of a flight?
  2. Worked exampleFind the maximum height of h(t) = -16t2 + 64t + 5 and the moment it happens.
  3. Worked exampleFind when h(t) = -16t2 + 48t brings the object back down to the ground.
  4. Common mistakeWhy isn't the largest allowed value of t the moment a ball is highest?
  5. Going furtherWhat changes about the arc when the launch speed is doubled?
Your turneasy1 of 3
A ball is launched upward from a platform 15 feet high at 94 feet per second, and gravity contributes -16t2 feet after t seconds. Write the height h(t) in feet, in terms of t.

Creating and Using an Exponential Function in Context

  1. How it worksHow does a percent change per period become the base of an exponential model?
  2. Worked exampleBuild a function for 400 bacteria tripling every 6 hours, then find the count at 24 hours.
  3. Worked exampleFind what 2000 dollars is worth after 5 years of losing 15 percent of its value annually.
  4. Common mistakeWhy isn't a 10 percent drop followed by a 10 percent rise a return to where you started?
  5. Going furtherWhat changes in the model when growth is compounded monthly rather than yearly?
Your turnhard1 of 3
A culture starts at 657 cells and multiplies by 4 every 11 hours. Write the number of cells as a function of the time t in hours.

Choosing Between a Quadratic and an Exponential Model

  1. How it worksHow does checking constant differences against constant ratios tell the two models apart?
  2. Worked exampleFind which model fits a table where y of 3, 6, 12, 24 lines up with x of 0, 1, 2, 3.
  3. Worked exampleFind which model fits a table pairing y of 1, 4, 9, 16 with x of 1, 2, 3, 4.
  4. Common mistakeWhy isn't every fast-rising curve an exponential one?
Your turneasy1 of 3
At x = 0, 1, 2 and 3 a table's outputs start at 7 and each one is 4 times the one before it. Which model fits?

Interpreting an Input-Output Pair, Intercept, or Initial Value

  1. How it worksHow does the multiplier in front of an exponential always land on the vertical intercept?
  2. Worked exampleExplain what the 850 and the 0.92 mean in V(t) = 850(0.92)t for a laptop's value.
  3. Common mistakeWhy isn't the base of an exponential model the percent change itself?
  4. Going furtherWhat does an input-output pair mean once the input goes negative in a time model?
Your turneasy1 of 3
A laptop's value in dollars after t years is V(t) = 1166(0.86)t. What do 1166 and 0.86 mean in this story?

Interpreting a Constant, Variable, Factor, or Term in Context

  1. How it worksHow does each factor of a quadratic model carry a specific moment in the story?
  2. Worked exampleExplain what the 2 and the 8 mean in h(x) = -5(x - 2)(x - 8) for an arch's height.
  3. Common mistakeWhy isn't the number multiplying a squared term the height of the curve?
  4. Going furtherWhat changes in the meaning when a leading coefficient flips from negative to positive?
Your turnmedium1 of 3
An arch's height in metres is h(x) = -7(x - 6)(x - 29), where x is the distance in metres from a marker. What do 6 and 29 describe?

Interpreting a Point on the Graph

  1. How it worksHow do you read a point's two coordinates back into the units of a story?
  2. Worked exampleExplain what the point (3, 180) means on a curve of revenue against price in dollars.
  3. Common mistakeWhy isn't the top of a revenue curve found at the highest price?
Your turneasy1 of 3
On a curve of a day's revenue R in dollars against the ticket price p in dollars, the point (29, 1368) lies on the curve. What does that point say?

Interpreting Stretches of a Graph, Not Just One Point

  1. How it worksHow does a flattening curve show a rate that is dying away?
  2. Worked exampleExplain what the rising half and the falling half of a projectile's height curve mean.
  3. Worked exampleFind the stretch where a downward parabola with roots at 1 and 9 is still increasing.
  4. Going furtherWhat does a curve levelling off towards a horizontal line mean in a cooling model?
Your turneasy1 of 3
A downward-opening parabola meets the x-axis at x = 6 and x = 23. Find the largest value of x at which it is still increasing.

Table, Rule, and Graph of a Quadratic or Exponential

  1. How it worksHow do second differences in a table reveal a quadratic hiding behind the numbers?
  2. Worked exampleFind the rule behind a table where x of 0, 1, 2, 3 gives y of 2, 6, 18, 54.
  3. Worked exampleFind the rule hiding behind a table pairing x of -1, 0, 1, 2 with y of 5, 0, -3, -4.
  4. Common mistakeWhy doesn't a table with a constant ratio ever come from a quadratic?
Your turnhard1 of 3
A quadratic has its vertex at (0, 10), and its table taken at unit steps in x has a constant second difference of 15. Write the rule for y in terms of x.

Graphs of Polynomial and Simple Rational Functions

  1. How it worksHow does each factor of a polynomial pin its graph to the horizontal axis at one spot?
  2. Worked exampleSketch y = (x + 2)(x - 1)(x - 3) from its three crossings and its end behaviour alone.
  3. Worked exampleFind where y = 6/(x - 4) blows up, and what the curve does far away from there.
  4. Common mistakeWhy doesn't a rational function's curve ever cross the vertical line it races towards?
Your turnhard1 of 3
A curve of the form y = n/(x - 15) passes through the point (0, -8). Write y in terms of x.

Transformations of a Nonlinear Graph

  1. How it worksHow does f(x - 3) + 2 slide a whole curve right 3 and up 2 without bending it?
  2. How it worksHow does a minus sign in front of a function flip it across the horizontal axis?
  3. Worked exampleShow me y = (x + 4)2 - 7 built as a shift of the basic parabola y = x2.
  4. Worked exampleFind the vertex of y = -2(x - 5)2 + 3 and which direction it opens.
  5. Common mistakeWhy isn't the graph of 2f(x) the same as the graph of f(2x)?
Your turneasy1 of 3
The parabola y = x2 is slid 7 units to the right and 8 units down. Write the new equation in terms of x.

Connecting a Table, a Rule, and a Curve in a Story

  1. How it worksHow do the units of a story cut down the inputs a nonlinear model is allowed?
  2. Worked exampleFind the sensible range of t for h(t) = -16t2 + 80t describing a ball thrown upward.
  3. Going furtherWhat happens to a growth model's curve when the data only covers whole months?
Your turnmedium1 of 3
A ball thrown upward from the ground has height h(t) = -16t2 + 121t feet after t seconds. Find the largest value of t that makes sense in the story.

The Most Useful Form of a Quadratic

  1. How it worksHow do standard, factored and vertex form each light up a different feature of the curve?
  2. Worked exampleShow me y = x2 - 6x + 8 in the form that displays its roots and the form that displays its vertex.
  3. Worked exampleFind the minimum value of y = x2 + 10x + 21 by moving to the form that displays it.
  4. Common mistakeWhy isn't the constant in standard form the lowest value the parabola reaches?
  5. Going furtherWhat feature does none of the three forms display without some extra work?
Your turnmedium1 of 3
Find the minimum value of y = x2 + 15x + 27.

The Most Useful Form of an Exponential

  1. How it worksHow does splitting an exponent turn a yearly rate into a monthly one?
  2. Worked exampleShow me 500(1.12)t rewritten to display the growth per quarter instead of per year.
  3. Worked exampleFind the yearly growth factor hidden inside P(t) = 300(1.05)4t.
  4. Common mistakeWhy isn't halving an exponent the same as halving the growth rate?
Your turnhard1 of 3
A balance is modelled by P(t) = 687(1.07)4t, with t in years. Find the yearly growth factor.

Function Notation and Evaluating

  1. How it worksHow does substituting a whole expression for x work the same way as substituting a number?
  2. Common mistakeWhy isn't f(x + 2) the same thing as f(x) + 2?
  3. Worked exampleFind f(-2) when f(x) = x2 - 4x + 1.
  4. Worked exampleFind g(a + 1) when g(x) = 2x2 - 3.
Your turnmedium1 of 3
For f(x) = x2 - 9x + 10, write f(-x) as a collected expression in x.

Finding the Input for a Given Output

  1. How it worksHow does setting a rule equal to the output turn the question into an equation?
  2. Worked exampleFind every x with f(x) = 12 when f(x) = x2 + x.
  3. Worked exampleFind t when 200 times 2 to the power t reaches 3200.
  4. Worked exampleFind x when the square root of (x - 1) equals 6.
  5. Common mistakeWhy doesn't every output value have an input somewhere that produces it?
Your turnhard1 of 3
For f(x) = the square root of (x - 9), write the input that produces an output of y, as an expression in y.

What the PSAT's Narrower Difficulty Band Means

  1. How it worksHow does one shared vertical scale let three tests with different ranges be compared directly?
  2. How it worksHow does a 320 to 1520 range sit inside a 400 to 1600 range as a staggered window?
  3. Worked exampleShow me one algebra skill written twice, once at PSAT difficulty and once at SAT difficulty.
  4. Common mistakeWhy isn't a PSAT score a loose prediction rather than a direct reading of the same scale?
Your turneasy1 of 3
A student scores 579 on PSAT/NMSQT math. What does that number mean beside an SAT math score of 579?

What the SAT Adds That the PSAT Leaves Out

  1. How it worksHow does the official framework drop rational-expression rewriting from PSAT scope but keep it for the SAT?
  2. Worked exampleShow me x3/4 rewritten in radical form, one of the rewrites reserved for the SAT.
  3. Worked exampleFind the solutions of (x - 2)(x + 5)(x - 1) = 0, the factored polynomial case the SAT adds.
  4. Going furtherWhat extra step does the SAT expect on simplifying (x2 - 9)/(x + 3) that the PSAT skips?
  5. Common mistakeWhy isn't a mention of circles on a PSAT math web page a reason to study them for it?
Your turnmedium1 of 3
A PSAT/NMSQT student has 43 days left and is choosing what to skip. Which Advanced Math skill is on the SAT but NOT on the PSAT?

AP®, SAT® and PSAT/NMSQT® are trademarks registered by the College Board (PSAT/NMSQT® jointly with the National Merit Scholarship Corporation). ACT® is a registered trademark of ACT Education Corp. GRE® is a registered trademark of ETS. LSAT® is a registered trademark of the Law School Admission Council. MCAT® is a registered trademark of the Association of American Medical Colleges. ISEE® and SSAT® are registered trademarks of their owners. None of these organizations is affiliated with, or endorses, Jentoo.