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.
Σχετικά με αυτήν τη δέσμη
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.
Κάρτες σε αυτήν τη δέσμη
Κάρτα 1
Ερώτηση
For real coefficients in ax² + bx + c = 0, with a ≠ 0, what is the discriminant?
Απάντηση
D = b² − 4ac. Its sign tells you the number of distinct real roots.
Κάρτα 2
Ερώτηση
Rewrite x² = 5x + 1 in zero form before using the discriminant.
Απάντηση
x² − 5x − 1 = 0. Use a = 1, b = −5 and c = −1.
Κάρτα 3
Ερώτηση
In −2x² + 7x − 4 = 0, what are a, b and c?
Απάντηση
a = −2, b = 7, c = −4. Keep the signs attached to the coefficients.
Κάρτα 4
Ερώτηση
Before using the quadratic discriminant on ax² + bx + c = 0, which coefficient must be nonzero?
Απάντηση
a ≠ 0. If a = 0, solve bx + c = 0 separately; the equation is no longer quadratic.
Κάρτα 5
Ερώτηση
A real quadratic has D > 0. How many distinct real roots?
Απάντηση
Two distinct real roots. The two square-root choices in the quadratic formula give different real values.
Κάρτα 6
Ερώτηση
Find the discriminant of 3x² − 2x − 4 = 0.
Απάντηση
52
D = (−2)² − 4(3)(−4) = 4 + 48 = 52.
Κάρτα 7
Ερώτηση
In 4x² + 9 = 0, what value of b belongs in D = b² − 4ac?
Απάντηση
b = 0. There is no x term; c = 9 is the constant coefficient.
Κάρτα 8
Ερώτηση
A real quadratic has a negative x² coefficient. Does that alone determine its number of real roots?
Απάντηση
No. The parabola opens downward, but D can be positive, zero or negative.
Κάρτα 9
Ερώτηση
A real quadratic has D = 0. How many distinct real roots?
Απάντηση
One distinct real root, repeated twice. Its value is −b/(2a).
Κάρτα 10
Ερώτηση
How many distinct real roots does 6x² + 5x − 6 = 0 have?
Απάντηση
Two. D = 5² − 4(6)(−6) = 169 > 0. You do not need to solve for the roots to count them.
Κάρτα 11
Ερώτηση
For b = −5, a student uses b² = −25 in the discriminant. What should replace −25?
Απάντηση
25
Square the whole signed coefficient: (−5)² = 25.
Κάρτα 12
Ερώτηση
For x² + 4x = 2, a student uses c = 2. What is the correct discriminant?
Απάντηση
24
First write x² + 4x − 2 = 0, so c = −2 and D = 4² − 4(1)(−2) = 24.
Κάρτα 13
Ερώτηση
A real quadratic has D < 0. How many real roots?
Απάντηση
None. A negative discriminant has no real square root.
Κάρτα 14
Ερώτηση
How many distinct real roots does 4x² + 12x + 9 = 0 have?
Απάντηση
One. D = 12² − 4(4)(9) = 0. The equation is (2x + 3)² = 0, so its repeated root is −3/2.
Κάρτα 15
Ερώτηση
How many distinct real roots does x² − 7x = 0 have?
Απάντηση
Two. D = (−7)² − 4(1)(0) = 49 > 0. Factoring gives x(x − 7) = 0, with roots 0 and 7.
Κάρτα 16
Ερώτηση
What discriminant condition is necessary and sufficient for a real quadratic to have two distinct real roots?
Απάντηση
D > 0. The condition assumes the x² coefficient is nonzero.
Κάρτα 17
Ερώτηση
How many real roots does 2x² + 4x + 5 = 0 have?
Απάντηση
None. D = 4² − 4(2)(5) = −24 < 0.
Κάρτα 18
Ερώτηση
A real quadratic graph touches the x-axis at exactly one point. What is its discriminant?
Απάντηση
D = 0. The touching point corresponds to a repeated real root.
Κάρτα 19
Ερώτηση
How many distinct real roots does −3x² + 2x + 2 = 0 have?
Απάντηση
Two. D = 2² − 4(−3)(2) = 28 > 0. A negative leading coefficient does not prevent real roots.
Κάρτα 20
Ερώτηση
For x² + 5 = 0, a student uses b = 5 and c = 0. What is the correct discriminant?
Απάντηση
−20. The coefficients are a = 1, b = 0 and c = 5, so D = 0² − 4(1)(5) = −20.
Κάρτα 21
Ερώτηση
What discriminant condition is necessary and sufficient for a real quadratic to have no real roots?
Απάντηση
D < 0. The coefficients are real and the x² coefficient is nonzero.
Κάρτα 22
Ερώτηση
For which real k does x² − 6x + k = 0 have a repeated root?
Απάντηση
k = 9. D = 36 − 4k = 0. At that value the equation is (x − 3)² = 0.
Κάρτα 23
Ερώτηση
Does D = 0 mean the root is zero?
Απάντηση
No. It means the two roots coincide. For (3x − 2)² = 0, the repeated root is 2/3, while D = 0.
Κάρτα 24
Ερώτηση
A real quadratic equation is multiplied throughout by a nonzero real λ. How does its discriminant change?
Απάντηση
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.
Κάρτα 25
Ερώτηση
An upward-opening real quadratic graph has its vertex strictly below the x-axis. What is the sign of D?
Απάντηση
Positive: D > 0. The graph crosses the x-axis twice. The vertex must be strictly below, not on, the axis.
Κάρτα 26
Ερώτηση
For which real k does x² + 8x + k = 0 have two distinct real roots?
Απάντηση
k < 16. D = 64 − 4k must be positive. At k = 16 there is only one distinct real root, so exclude the boundary.
Κάρτα 27
Ερώτηση
How many distinct real roots does −2x² + 8x − 8 = 0 have?
Απάντηση
One. D = 8² − 4(−2)(−8) = 0. The equation is −2(x − 2)² = 0, so the repeated root is 2.
Κάρτα 28
Ερώτηση
For which real k is (k − 2)x² + 3x + 1 = 0 a quadratic equation?
Απάντηση
k ≠ 2. The x² coefficient is k − 2, which must be nonzero.
Κάρτα 29
Ερώτηση
For which real t does x² + 2tx + 9 = 0 have no real roots?
Απάντηση
−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.
Κάρτα 30
Ερώτηση
For which real k does x² + 10x + k = 0 have no real roots?
Απάντηση
k > 25. D = 100 − 4k must be negative. At k = 25 the repeated root is −5, so exclude that boundary.
Κάρτα 31
Ερώτηση
For which real k does x² + kx − 4 = 0 have two distinct real roots?
Απάντηση
Every real k. D = k² + 16 is always positive, and the leading coefficient stays 1.
Κάρτα 32
Ερώτηση
At k = 2, how many real solutions does (k − 2)x² + 3x + 1 = 0 have?
Απάντηση
One: x = −1/3. At k = 2 the x² term vanishes and the equation becomes 3x + 1 = 0; use the linear equation.
Κάρτα 33
Ερώτηση
For which real k does kx² + 2x + 1 = 0 have exactly one distinct real solution?
Απάντηση
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.
Κάρτα 34
Ερώτηση
For which real t does x² + 2tx + 9 = 0 have two distinct real roots?
Απάντηση
t < −3 or t > 3. D = 4t² − 36 > 0 gives t² > 9. Both boundary values give a repeated root and are excluded.
Κάρτα 35
Ερώτηση
For which real k does kx² + 6x + 2 = 0 have no real solutions?
Απάντηση
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.
Κάρτα 36
Ερώτηση
At k = 4, how many real solutions does (k − 4)x² + (k − 4)x + 1 = 0 have?
Απάντηση
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.
Κάρτα 37
Ερώτηση
At k = 0, how many real solutions does kx² + kx = 0 have?
Απάντηση
Infinitely many: every real x. The equation becomes 0 = 0. Do not apply the quadratic D = 0 rule when the leading coefficient is zero.
Κάρτα 38
Ερώτηση
How many distinct real roots does x² = 0 have?
Απάντηση
One: x = 0. D = 0; zero is a repeated root, counted once when counting distinct roots.
Κάρτα 39
Ερώτηση
Do x² − 2x − 2 = 0 and −3x² + 6x + 6 = 0 have different numbers of distinct real roots?
Απάντηση
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.
Κάρτα 40
Ερώτηση
For which real k does (k − 4)x² + (k − 4)x + 1 = 0 have two distinct real roots?
Απάντηση
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.
40 κάρτες
Discriminant Flashcards: Real Roots & Parameter Boundaries
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