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Discriminant Flashcards: Real Roots & Parameter Boundaries

Practice 40 original discriminant prompts: count distinct real roots, repair coefficient errors, interpret graph clues, and check parameter boundaries when the x² term vanishes.

Par Open Study Contributor

À propos de ce paquet

Count distinct real roots without solving every quadratic. These 40 discriminant flashcards give English-reading algebra learners original short decisions about signed coefficients, repeated roots, graph clues and parameter boundaries.

Practice runs in several directions: equation or discriminant sign → real-root count; a stated root count or graph position → discriminant condition; and a short parameter equation → the values that meet a root-count condition. Error-repair cards address missing coefficients, signs, zero form, and the difference between a zero discriminant and a zero root. Later cards check when the x² term vanishes: the remaining equation may be linear, a contradiction or an identity. Every prompt carries the context needed for mixed review.

The sequence introduces zero form, coefficient reading and the nonzero leading-coefficient requirement, then adds the three sign cases with short calculations between them. Reverse cues and error diagnosis lead into parameter intervals and degenerate cases. Related variants are separated by at least three other cards. All cards use the discriminant tag; the review scheduler handles long-term spacing.

Know signed arithmetic, squares and simple inequalities before starting. Coefficients and parameters are real unless a card says otherwise, and root counts mean distinct real roots. This evergreen Mathematics deck has no exam or curriculum alignment. It excludes complex-root calculations, rational-versus-irrational classification, higher-degree discriminants, full quadratic-formula drills and long proofs. Arbitrary reversals from a root count to unique coefficients are excluded because that information does not determine a unique equation.

The discriminant identity and quadratic-root formula were fact-checked using NIST DLMF §1.11. All examples, wording and organization were independently created from common algebra and checked with exact arithmetic, completing-square reasoning, factorization or substitution. No textbook exercises, competitor cards, exam questions, source prose or diagrams were copied or adapted.

AI tools assisted with drafting and the original decorative cover. Every final front and back received author review, with an independent readiness review before publication; this does not claim learner testing. The CC0 dedication covers original wording, invented examples, organization, metadata and generated cover to the extent applicable rights exist. Mathematical facts and cited materials retain their own status. This is independent study practice, with no affiliation to or endorsement by NIST or any course or examination provider.

Cartes de ce paquet

  1. Carte 1

    Question

    For real coefficients in ax² + bx + c = 0, with a ≠ 0, what is the discriminant?

    Réponse

    D = b² − 4ac. Its sign tells you the number of distinct real roots.

  2. Carte 2

    Question

    Rewrite x² = 5x + 1 in zero form before using the discriminant.

    Réponse

    x² − 5x − 1 = 0. Use a = 1, b = −5 and c = −1.

  3. Carte 3

    Question

    In −2x² + 7x − 4 = 0, what are a, b and c?

    Réponse

    a = −2, b = 7, c = −4. Keep the signs attached to the coefficients.

  4. Carte 4

    Question

    Before using the quadratic discriminant on ax² + bx + c = 0, which coefficient must be nonzero?

    Réponse

    a ≠ 0. If a = 0, solve bx + c = 0 separately; the equation is no longer quadratic.

  5. Carte 5

    Question

    A real quadratic has D > 0. How many distinct real roots?

    Réponse

    Two distinct real roots. The two square-root choices in the quadratic formula give different real values.

  6. Carte 6

    Question

    Find the discriminant of 3x² − 2x − 4 = 0.

    Réponse

    52

    D = (−2)² − 4(3)(−4) = 4 + 48 = 52.

  7. Carte 7

    Question

    In 4x² + 9 = 0, what value of b belongs in D = b² − 4ac?

    Réponse

    b = 0. There is no x term; c = 9 is the constant coefficient.

  8. Carte 8

    Question

    A real quadratic has a negative x² coefficient. Does that alone determine its number of real roots?

    Réponse

    No. The parabola opens downward, but D can be positive, zero or negative.

  9. Carte 9

    Question

    A real quadratic has D = 0. How many distinct real roots?

    Réponse

    One distinct real root, repeated twice. Its value is −b/(2a).

  10. Carte 10

    Question

    How many distinct real roots does 6x² + 5x − 6 = 0 have?

    Réponse

    Two. D = 5² − 4(6)(−6) = 169 > 0. You do not need to solve for the roots to count them.

  11. Carte 11

    Question

    For b = −5, a student uses b² = −25 in the discriminant. What should replace −25?

    Réponse

    25

    Square the whole signed coefficient: (−5)² = 25.

  12. Carte 12

    Question

    For x² + 4x = 2, a student uses c = 2. What is the correct discriminant?

    Réponse

    24

    First write x² + 4x − 2 = 0, so c = −2 and D = 4² − 4(1)(−2) = 24.

  13. Carte 13

    Question

    A real quadratic has D < 0. How many real roots?

    Réponse

    None. A negative discriminant has no real square root.

  14. Carte 14

    Question

    How many distinct real roots does 4x² + 12x + 9 = 0 have?

    Réponse

    One. D = 12² − 4(4)(9) = 0. The equation is (2x + 3)² = 0, so its repeated root is −3/2.

  15. Carte 15

    Question

    How many distinct real roots does x² − 7x = 0 have?

    Réponse

    Two. D = (−7)² − 4(1)(0) = 49 > 0. Factoring gives x(x − 7) = 0, with roots 0 and 7.

  16. Carte 16

    Question

    What discriminant condition is necessary and sufficient for a real quadratic to have two distinct real roots?

    Réponse

    D > 0. The condition assumes the x² coefficient is nonzero.

  17. Carte 17

    Question

    How many real roots does 2x² + 4x + 5 = 0 have?

    Réponse

    None. D = 4² − 4(2)(5) = −24 < 0.

  18. Carte 18

    Question

    A real quadratic graph touches the x-axis at exactly one point. What is its discriminant?

    Réponse

    D = 0. The touching point corresponds to a repeated real root.

  19. Carte 19

    Question

    How many distinct real roots does −3x² + 2x + 2 = 0 have?

    Réponse

    Two. D = 2² − 4(−3)(2) = 28 > 0. A negative leading coefficient does not prevent real roots.

  20. Carte 20

    Question

    For x² + 5 = 0, a student uses b = 5 and c = 0. What is the correct discriminant?

    Réponse

    −20. The coefficients are a = 1, b = 0 and c = 5, so D = 0² − 4(1)(5) = −20.

  21. Carte 21

    Question

    What discriminant condition is necessary and sufficient for a real quadratic to have no real roots?

    Réponse

    D < 0. The coefficients are real and the x² coefficient is nonzero.

  22. Carte 22

    Question

    For which real k does x² − 6x + k = 0 have a repeated root?

    Réponse

    k = 9. D = 36 − 4k = 0. At that value the equation is (x − 3)² = 0.

  23. Carte 23

    Question

    Does D = 0 mean the root is zero?

    Réponse

    No. It means the two roots coincide. For (3x − 2)² = 0, the repeated root is 2/3, while D = 0.

  24. Carte 24

    Question

    A real quadratic equation is multiplied throughout by a nonzero real λ. How does its discriminant change?

    Réponse

    It becomes λ²D. Since λ² > 0, the sign and number of distinct real roots stay the same. Multiplying by zero would turn the equation into 0 = 0.

  25. Carte 25

    Question

    An upward-opening real quadratic graph has its vertex strictly below the x-axis. What is the sign of D?

    Réponse

    Positive: D > 0. The graph crosses the x-axis twice. The vertex must be strictly below, not on, the axis.

  26. Carte 26

    Question

    For which real k does x² + 8x + k = 0 have two distinct real roots?

    Réponse

    k < 16. D = 64 − 4k must be positive. At k = 16 there is only one distinct real root, so exclude the boundary.

  27. Carte 27

    Question

    How many distinct real roots does −2x² + 8x − 8 = 0 have?

    Réponse

    One. D = 8² − 4(−2)(−8) = 0. The equation is −2(x − 2)² = 0, so the repeated root is 2.

  28. Carte 28

    Question

    For which real k is (k − 2)x² + 3x + 1 = 0 a quadratic equation?

    Réponse

    k ≠ 2. The x² coefficient is k − 2, which must be nonzero.

  29. Carte 29

    Question

    For which real t does x² + 2tx + 9 = 0 have no real roots?

    Réponse

    −3 < t < 3. D = 4t² − 36 < 0 gives t² < 9. At t = −3 or 3 there is a repeated real root, so exclude both endpoints.

  30. Carte 30

    Question

    For which real k does x² + 10x + k = 0 have no real roots?

    Réponse

    k > 25. D = 100 − 4k must be negative. At k = 25 the repeated root is −5, so exclude that boundary.

  31. Carte 31

    Question

    For which real k does x² + kx − 4 = 0 have two distinct real roots?

    Réponse

    Every real k. D = k² + 16 is always positive, and the leading coefficient stays 1.

  32. Carte 32

    Question

    At k = 2, how many real solutions does (k − 2)x² + 3x + 1 = 0 have?

    Réponse

    One: x = −1/3. At k = 2 the x² term vanishes and the equation becomes 3x + 1 = 0; use the linear equation.

  33. Carte 33

    Question

    For which real k does kx² + 2x + 1 = 0 have exactly one distinct real solution?

    Réponse

    k = 0 or k = 1. At k = 0 the equation is linear, with x = −1/2. For k ≠ 0, D = 4 − 4k must be zero, giving k = 1 and repeated root −1.

  34. Carte 34

    Question

    For which real t does x² + 2tx + 9 = 0 have two distinct real roots?

    Réponse

    t < −3 or t > 3. D = 4t² − 36 > 0 gives t² > 9. Both boundary values give a repeated root and are excluded.

  35. Carte 35

    Question

    For which real k does kx² + 6x + 2 = 0 have no real solutions?

    Réponse

    k > 9/2. For k ≠ 0, D = 36 − 8k < 0 gives this interval. At k = 0 the linear equation 6x + 2 = 0 has one solution; at k = 9/2 there is a repeated root.

  36. Carte 36

    Question

    At k = 4, how many real solutions does (k − 4)x² + (k − 4)x + 1 = 0 have?

    Réponse

    None. Both variable terms vanish, leaving 1 = 0, which is false for every x. A zero leading coefficient does not always leave a solvable linear equation.

  37. Carte 37

    Question

    At k = 0, how many real solutions does kx² + kx = 0 have?

    Réponse

    Infinitely many: every real x. The equation becomes 0 = 0. Do not apply the quadratic D = 0 rule when the leading coefficient is zero.

  38. Carte 38

    Question

    How many distinct real roots does x² = 0 have?

    Réponse

    One: x = 0. D = 0; zero is a repeated root, counted once when counting distinct roots.

  39. Carte 39

    Question

    Do x² − 2x − 2 = 0 and −3x² + 6x + 6 = 0 have different numbers of distinct real roots?

    Réponse

    No; each has two. Their discriminants are 12 and 108, both positive. The second equation is the first multiplied by −3, so even the roots themselves are unchanged.

  40. Carte 40

    Question

    For which real k does (k − 4)x² + (k − 4)x + 1 = 0 have two distinct real roots?

    Réponse

    k < 4 or k > 8. For k ≠ 4, D = (k − 4)(k − 8) must be positive. At k = 4 the equation is 1 = 0; at k = 8 it has a repeated root. Exclude both boundaries.

An orange upward-opening parabola crossing a blue horizontal axis at two pale points on a dark background.

40 cartes

Discriminant Flashcards: Real Roots & Parameter Boundaries

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