Unknown-constant and solution-count problems — 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.

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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.

Question 1

Guide question E4.1

For what value of k does kx + 4 = 5x + 4 have infinitely many real solutions?

Your answer

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

Why this answer

Rearrange to $(k - 5)x = 0.$ At $k = 5,$ the equation is $0 = 0,$ true for every real x. At any other k, only $x = 0$ works. The question asks for infinitely many solutions, so $k = 5.$ Dividing by $k - 5$ before examining whether it is zero would discard the desired case. See Example 4.01.

Review lesson: Unknown-constant and solution-count problems

Question 2

Guide question E4.2

For what value of c does x² − 10x + c = 0 have exactly one real solution?

Your answer

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

Why this answer

The equation is genuinely quadratic for every c. Exactly one distinct real solution requires discriminant $(-10) 2 - 4(1)c = 0,$ so $100 - 4c = 0$ and $c = 25.$ Check: $x 2 - 10x + 25 = (x - 5) 2 .$ The repeated root counts as one distinct real solution. See Example 4.06.

Review lesson: Unknown-constant and solution-count problems

Question 3

Guide question E4.3

For what value of k does the system 3x + 2y = 7, 9x + ky = 25 have no solution?

Your answer

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

Why this answer

Multiplying the first equation by 3 gives $9x + 6y = 21.$ At $k = 6,$ the second equation demands the same left side equal 25, an impossibility. Thus the lines are parallel and distinct, and the system has no solution. Matching coefficients alone is not enough for infinitely many solutions; the constant terms must match too. See Example 4.03.

Review lesson: Unknown-constant and solution-count problems

Question 5

Guide question E4.5

For what value of k does (x² − 16)/(x − 4) = k have no real solution?

Your answer

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

Why this answer

The original denominator excludes $x = 4.$ For every allowed input, the left side simplifies to $x + 4,$ so the candidate is $x = k - 4.$ This candidate is excluded precisely when $k - 4 = 4,$ or $k = 8.$ At every other k it is valid. The simplified line must retain its missing input. See Example 4.12.

Review lesson: Unknown-constant and solution-count problems

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Written self-checks

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

Guide question E4.4

Find every real k for which (k − 2)x² + 2x + 1 = 0 has exactly one real solution in x.

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$k = 2$ or $k = 3$

First test the reduced-degree case: $k = 2$ gives $2x + 1 = 0,$ exactly one solution. For $k \neq 2,$ use the quadratic discriminant: $4 - 4(k - 2) = 12 - 4k = 0,$ so $k = 3.$ At that value the equation is $(x + 1) 2 = 0.$ These two cases exhaust the possibilities. See Example 4.11.

Practice explanations

The drill

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Why this answer

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Review practice

Written self-checks

Guide question E4.4