Open Practice Tests

Plugging in values and testing answer choices — Topic practice

5 exercises · 4 automatically checked · 1 written self-check · Untimed

Instructions and review guidance

4 questions · Shuffled order · Untimed practice

Try each question before choosing Check answer. Automatic checks assess your final numeric or choice answer only; compare your method, model and checks with the full explanation. Guide question labels stay the same when you shuffle the bank. 1 written response appears below the bank for self-review.

Guide question labels stay the same when the order changes. Check answers when you are ready to review.

Written self-checks are below the question bank and are excluded from the automatic score.

Written responses are self-review exercises.

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Practice explanations

Use these explanations to diagnose the first wrong decision, not only to compare final answers. Each solution appears with its question and links to a worked example with a transferable method. A different valid solution is welcome; compare its length, assumptions, and reliability.

The drill

Question 1

  • Guide question E2.1

For all real x, which expression equals (x + 2)2 − x2?

Question 2

  • Guide question E2.2

After a 15% discount, an item costs $68. What was the original price?

Question 3

  • Guide question E2.3

Positive a and b satisfy a = 4b. Find (a + b)/(a − b).

Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.

Question 5

  • Guide question E2.5

The identity (x − 3)(ax + b) = 2x2 − 5x − 3 holds for all real x. Find a + b without first finding both constants separately.

Enter an integer, decimal or fraction without units. Use up to 5 characters, or 6 including a leading minus.

Question navigator and options
  • Answered
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  • Incorrect
  • Revealed, not answered
  • Current

Written self-checks

Write your complete response before opening the model answer. Compare both your answer and your reasoning.

Guide question E2.4

A student tests x = 1 to decide whether x2 = x holds for every real x. The test succeeds. Give an allowed counterexample, and state what it proves.

Show model answerHide model answer for question E2.4

For example, x = 2

At x = 2, the claimed equality becomes 4 = 2, which is false. One valid counterexample disproves a statement asserted for every real x. Testing x = 1 only shows that the equality holds at that input; it does not establish an identity. In fact, x2 = x holds only when x(x − 1) = 0, so only at 0 and 1. See Example 2.12.