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