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