Polynomials and quadratic equations — ISEE Upper practice questions

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Practice 76 Polynomials and quadratic equations questions in the app

What this topic is

Factoring, completing the square and setting a quadratic equal to zero are the main tools for Polynomials and quadratic equations.

Completing the square rests on one fact: adding the square of half the x coefficient makes a perfect square trinomial, so x squared plus six x becomes x plus three, squared, once nine is added. Pattern questions sometimes describe two quantities growing at different rates and ask when they match, which leads to a quadratic with one sensible positive root.

A well-known misconception is dividing both sides by x and losing a solution. Conditions like x being a positive integer often appear in quantitative comparisons.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Easier

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

4✗ Not this one5✓ Correct6✗ Not this one10✗ Not this one

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Answer: B — 5

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

Question 2Mid

$x$ is a positive integer.

Column A	Column B
$(x + 4) (x - 4)$	$\frac{x^{3} - 16 x}{x}$

The quantity in Column A is greater✗ Not this oneThe quantity in Column B is greater✗ Not this oneThe two quantities are equal✓ CorrectThe relationship cannot be determined from the information given✗ Not this one

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Answer: C — The two quantities are equal

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

Question 3Easier

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

5✗ Not this one10✗ Not this one20✗ Not this one25✓ Correct

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Answer: D — 25

Completing the square uses ${(\frac{10}{2})}^{2} = 25$ , and $x^{2} + 10 x + 25 = (x + 5)^{2}$ .

Every question is picked for where you are right now, wrong answers come back until they stick, and it all works offline.

Practice 76 Polynomials and quadratic equations questions in the app

Polynomials and quadratic equations — ISEE Upper practice questions

What this topic is

Sample questions

More ISEE Upper topics

Quantitative Reasoning