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