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.
دربارهٔ این دسته
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.
کارت‌های این دسته
کارت ۱
پرسش
For real coefficients in ax² + bx + c = 0, with a ≠ 0, what is the discriminant?
پاسخ
D = b² − 4ac. Its sign tells you the number of distinct real roots.
کارت ۲
پرسش
Rewrite x² = 5x + 1 in zero form before using the discriminant.
پاسخ
x² − 5x − 1 = 0. Use a = 1, b = −5 and c = −1.
کارت ۳
پرسش
In −2x² + 7x − 4 = 0, what are a, b and c?
پاسخ
a = −2, b = 7, c = −4. Keep the signs attached to the coefficients.
کارت ۴
پرسش
Before using the quadratic discriminant on ax² + bx + c = 0, which coefficient must be nonzero?
پاسخ
a ≠ 0. If a = 0, solve bx + c = 0 separately; the equation is no longer quadratic.
کارت ۵
پرسش
A real quadratic has D > 0. How many distinct real roots?
پاسخ
Two distinct real roots. The two square-root choices in the quadratic formula give different real values.
کارت ۶
پرسش
Find the discriminant of 3x² − 2x − 4 = 0.
پاسخ
52
D = (−2)² − 4(3)(−4) = 4 + 48 = 52.
کارت ۷
پرسش
In 4x² + 9 = 0, what value of b belongs in D = b² − 4ac?
پاسخ
b = 0. There is no x term; c = 9 is the constant coefficient.
کارت ۸
پرسش
A real quadratic has a negative x² coefficient. Does that alone determine its number of real roots?
پاسخ
No. The parabola opens downward, but D can be positive, zero or negative.
کارت ۹
پرسش
A real quadratic has D = 0. How many distinct real roots?
پاسخ
One distinct real root, repeated twice. Its value is −b/(2a).
کارت ۱۰
پرسش
How many distinct real roots does 6x² + 5x − 6 = 0 have?
پاسخ
Two. D = 5² − 4(6)(−6) = 169 > 0. You do not need to solve for the roots to count them.
کارت ۱۱
پرسش
For b = −5, a student uses b² = −25 in the discriminant. What should replace −25?
پاسخ
25
Square the whole signed coefficient: (−5)² = 25.
کارت ۱۲
پرسش
For x² + 4x = 2, a student uses c = 2. What is the correct discriminant?
پاسخ
24
First write x² + 4x − 2 = 0, so c = −2 and D = 4² − 4(1)(−2) = 24.
کارت ۱۳
پرسش
A real quadratic has D < 0. How many real roots?
پاسخ
None. A negative discriminant has no real square root.
کارت ۱۴
پرسش
How many distinct real roots does 4x² + 12x + 9 = 0 have?
پاسخ
One. D = 12² − 4(4)(9) = 0. The equation is (2x + 3)² = 0, so its repeated root is −3/2.
کارت ۱۵
پرسش
How many distinct real roots does x² − 7x = 0 have?
پاسخ
Two. D = (−7)² − 4(1)(0) = 49 > 0. Factoring gives x(x − 7) = 0, with roots 0 and 7.
کارت ۱۶
پرسش
What discriminant condition is necessary and sufficient for a real quadratic to have two distinct real roots?
پاسخ
D > 0. The condition assumes the x² coefficient is nonzero.
کارت ۱۷
پرسش
How many real roots does 2x² + 4x + 5 = 0 have?
پاسخ
None. D = 4² − 4(2)(5) = −24 < 0.
کارت ۱۸
پرسش
A real quadratic graph touches the x-axis at exactly one point. What is its discriminant?
پاسخ
D = 0. The touching point corresponds to a repeated real root.
کارت ۱۹
پرسش
How many distinct real roots does −3x² + 2x + 2 = 0 have?
پاسخ
Two. D = 2² − 4(−3)(2) = 28 > 0. A negative leading coefficient does not prevent real roots.
کارت ۲۰
پرسش
For x² + 5 = 0, a student uses b = 5 and c = 0. What is the correct discriminant?
پاسخ
−20. The coefficients are a = 1, b = 0 and c = 5, so D = 0² − 4(1)(5) = −20.
کارت ۲۱
پرسش
What discriminant condition is necessary and sufficient for a real quadratic to have no real roots?
پاسخ
D < 0. The coefficients are real and the x² coefficient is nonzero.
کارت ۲۲
پرسش
For which real k does x² − 6x + k = 0 have a repeated root?
پاسخ
k = 9. D = 36 − 4k = 0. At that value the equation is (x − 3)² = 0.
کارت ۲۳
پرسش
Does D = 0 mean the root is zero?
پاسخ
No. It means the two roots coincide. For (3x − 2)² = 0, the repeated root is 2/3, while D = 0.
کارت ۲۴
پرسش
A real quadratic equation is multiplied throughout by a nonzero real λ. How does its discriminant change?
پاسخ
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.
کارت ۲۵
پرسش
An upward-opening real quadratic graph has its vertex strictly below the x-axis. What is the sign of D?
پاسخ
Positive: D > 0. The graph crosses the x-axis twice. The vertex must be strictly below, not on, the axis.
کارت ۲۶
پرسش
For which real k does x² + 8x + k = 0 have two distinct real roots?
پاسخ
k < 16. D = 64 − 4k must be positive. At k = 16 there is only one distinct real root, so exclude the boundary.
کارت ۲۷
پرسش
How many distinct real roots does −2x² + 8x − 8 = 0 have?
پاسخ
One. D = 8² − 4(−2)(−8) = 0. The equation is −2(x − 2)² = 0, so the repeated root is 2.
کارت ۲۸
پرسش
For which real k is (k − 2)x² + 3x + 1 = 0 a quadratic equation?
پاسخ
k ≠ 2. The x² coefficient is k − 2, which must be nonzero.
کارت ۲۹
پرسش
For which real t does x² + 2tx + 9 = 0 have no real roots?
پاسخ
−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.
کارت ۳۰
پرسش
For which real k does x² + 10x + k = 0 have no real roots?
پاسخ
k > 25. D = 100 − 4k must be negative. At k = 25 the repeated root is −5, so exclude that boundary.
کارت ۳۱
پرسش
For which real k does x² + kx − 4 = 0 have two distinct real roots?
پاسخ
Every real k. D = k² + 16 is always positive, and the leading coefficient stays 1.
کارت ۳۲
پرسش
At k = 2, how many real solutions does (k − 2)x² + 3x + 1 = 0 have?
پاسخ
One: x = −1/3. At k = 2 the x² term vanishes and the equation becomes 3x + 1 = 0; use the linear equation.
کارت ۳۳
پرسش
For which real k does kx² + 2x + 1 = 0 have exactly one distinct real solution?
پاسخ
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.
کارت ۳۴
پرسش
For which real t does x² + 2tx + 9 = 0 have two distinct real roots?
پاسخ
t < −3 or t > 3. D = 4t² − 36 > 0 gives t² > 9. Both boundary values give a repeated root and are excluded.
کارت ۳۵
پرسش
For which real k does kx² + 6x + 2 = 0 have no real solutions?
پاسخ
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.
کارت ۳۶
پرسش
At k = 4, how many real solutions does (k − 4)x² + (k − 4)x + 1 = 0 have?
پاسخ
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.
کارت ۳۷
پرسش
At k = 0, how many real solutions does kx² + kx = 0 have?
پاسخ
Infinitely many: every real x. The equation becomes 0 = 0. Do not apply the quadratic D = 0 rule when the leading coefficient is zero.
کارت ۳۸
پرسش
How many distinct real roots does x² = 0 have?
پاسخ
One: x = 0. D = 0; zero is a repeated root, counted once when counting distinct roots.
کارت ۳۹
پرسش
Do x² − 2x − 2 = 0 and −3x² + 6x + 6 = 0 have different numbers of distinct real roots?
پاسخ
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.
کارت ۴۰
پرسش
For which real k does (k − 4)x² + (k − 4)x + 1 = 0 have two distinct real roots?
پاسخ
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.
۴۰ کارت
Discriminant Flashcards: Real Roots & Parameter Boundaries
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