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Quantitative ReasoningFree

Polynomials and quadratic equations — ISEE Upper practice questions

On the ISEE Upper Quantitative Reasoning section, Polynomials and quadratic equations covers degree-two expressions: expanding, factoring, completing the square, and solving. A student must set polynomial expressions equal, form a perfect square trinomial, and honor constraints such as a positive integer variable. Items are typically short word problems or direct algebra questions whose choices are nearby integers or plausible constants. Traps include dropping a domain restriction, adding the wrong constant to complete the square, and assuming a linear and a quadratic expression match for every value instead of one valid case.

  • Section: Quantitative Reasoning 37 questions on the paper
  • 74 practice questions in the app

Sample questions

Question 1

In a tile pattern, Figure nn has n2n^2 white tiles and 3n+103n+10 gray tiles. For which value of nn does the figure have the same number of white and gray tiles?

  1. A4
  2. B5
  3. C6
  4. D10
Show answer

Answer: B — 5

The counts are equal when n2=3n+10n^2=3n+10. Testing the choices, n=5n=5 gives 25 white tiles and 3(5)+10=253(5)+10=25 gray tiles.

Question 2

xx is a positive integer.

Column AColumn B
(x+4)(x−4)(x+4)(x-4)x3−16xx\dfrac{x^3-16x}{x}
  1. AThe quantity in Column A is greater
  2. BThe quantity in Column B is greater
  3. CThe two quantities are equal
  4. DThe relationship cannot be determined from the information given
Show answer

Answer: C — The two quantities are equal

Column A expands to x2−16x^2-16. In Column B, dividing each term by xx gives x2−16x^2-16 as well (valid since x>0x>0), so the two are equal.

Question 3

Which constant should be added to x2+10xx^2+10x so that the result is a perfect square trinomial?

  1. A5
  2. B10
  3. C20
  4. D25
Show answer

Answer: D — 25

Completing the square uses (102)2=25\left(\dfrac{10}{2}\right)^2=25, and x2+10x+25=(x+5)2x^2+10x+25=(x+5)^2.

Practice 74 Polynomials and quadratic equations questions in the app

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