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