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.
এই ডেক সম্পর্কে
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.
এই ডেকের কার্ড
কার্ড ১
প্রশ্ন
For real coefficients in ax² + bx + c = 0, with a ≠ 0, what is the discriminant?
উত্তর
D = b² − 4ac. Its sign tells you the number of distinct real roots.
কার্ড ২
প্রশ্ন
Rewrite x² = 5x + 1 in zero form before using the discriminant.
উত্তর
x² − 5x − 1 = 0. Use a = 1, b = −5 and c = −1.
কার্ড ৩
প্রশ্ন
In −2x² + 7x − 4 = 0, what are a, b and c?
উত্তর
a = −2, b = 7, c = −4. Keep the signs attached to the coefficients.
কার্ড ৪
প্রশ্ন
Before using the quadratic discriminant on ax² + bx + c = 0, which coefficient must be nonzero?
উত্তর
a ≠ 0. If a = 0, solve bx + c = 0 separately; the equation is no longer quadratic.
কার্ড ৫
প্রশ্ন
A real quadratic has D > 0. How many distinct real roots?
উত্তর
Two distinct real roots. The two square-root choices in the quadratic formula give different real values.
কার্ড ৬
প্রশ্ন
Find the discriminant of 3x² − 2x − 4 = 0.
উত্তর
52
D = (−2)² − 4(3)(−4) = 4 + 48 = 52.
কার্ড ৭
প্রশ্ন
In 4x² + 9 = 0, what value of b belongs in D = b² − 4ac?
উত্তর
b = 0. There is no x term; c = 9 is the constant coefficient.
কার্ড ৮
প্রশ্ন
A real quadratic has a negative x² coefficient. Does that alone determine its number of real roots?
উত্তর
No. The parabola opens downward, but D can be positive, zero or negative.
কার্ড ৯
প্রশ্ন
A real quadratic has D = 0. How many distinct real roots?
উত্তর
One distinct real root, repeated twice. Its value is −b/(2a).
কার্ড ১০
প্রশ্ন
How many distinct real roots does 6x² + 5x − 6 = 0 have?
উত্তর
Two. D = 5² − 4(6)(−6) = 169 > 0. You do not need to solve for the roots to count them.
কার্ড ১১
প্রশ্ন
For b = −5, a student uses b² = −25 in the discriminant. What should replace −25?
উত্তর
25
Square the whole signed coefficient: (−5)² = 25.
কার্ড ১২
প্রশ্ন
For x² + 4x = 2, a student uses c = 2. What is the correct discriminant?
উত্তর
24
First write x² + 4x − 2 = 0, so c = −2 and D = 4² − 4(1)(−2) = 24.
কার্ড ১৩
প্রশ্ন
A real quadratic has D < 0. How many real roots?
উত্তর
None. A negative discriminant has no real square root.
কার্ড ১৪
প্রশ্ন
How many distinct real roots does 4x² + 12x + 9 = 0 have?
উত্তর
One. D = 12² − 4(4)(9) = 0. The equation is (2x + 3)² = 0, so its repeated root is −3/2.
কার্ড ১৫
প্রশ্ন
How many distinct real roots does x² − 7x = 0 have?
উত্তর
Two. D = (−7)² − 4(1)(0) = 49 > 0. Factoring gives x(x − 7) = 0, with roots 0 and 7.
কার্ড ১৬
প্রশ্ন
What discriminant condition is necessary and sufficient for a real quadratic to have two distinct real roots?
উত্তর
D > 0. The condition assumes the x² coefficient is nonzero.
কার্ড ১৭
প্রশ্ন
How many real roots does 2x² + 4x + 5 = 0 have?
উত্তর
None. D = 4² − 4(2)(5) = −24 < 0.
কার্ড ১৮
প্রশ্ন
A real quadratic graph touches the x-axis at exactly one point. What is its discriminant?
উত্তর
D = 0. The touching point corresponds to a repeated real root.
কার্ড ১৯
প্রশ্ন
How many distinct real roots does −3x² + 2x + 2 = 0 have?
উত্তর
Two. D = 2² − 4(−3)(2) = 28 > 0. A negative leading coefficient does not prevent real roots.
কার্ড ২০
প্রশ্ন
For x² + 5 = 0, a student uses b = 5 and c = 0. What is the correct discriminant?
উত্তর
−20. The coefficients are a = 1, b = 0 and c = 5, so D = 0² − 4(1)(5) = −20.
কার্ড ২১
প্রশ্ন
What discriminant condition is necessary and sufficient for a real quadratic to have no real roots?
উত্তর
D < 0. The coefficients are real and the x² coefficient is nonzero.
কার্ড ২২
প্রশ্ন
For which real k does x² − 6x + k = 0 have a repeated root?
উত্তর
k = 9. D = 36 − 4k = 0. At that value the equation is (x − 3)² = 0.
কার্ড ২৩
প্রশ্ন
Does D = 0 mean the root is zero?
উত্তর
No. It means the two roots coincide. For (3x − 2)² = 0, the repeated root is 2/3, while D = 0.
কার্ড ২৪
প্রশ্ন
A real quadratic equation is multiplied throughout by a nonzero real λ. How does its discriminant change?
উত্তর
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.
কার্ড ২৫
প্রশ্ন
An upward-opening real quadratic graph has its vertex strictly below the x-axis. What is the sign of D?
উত্তর
Positive: D > 0. The graph crosses the x-axis twice. The vertex must be strictly below, not on, the axis.
কার্ড ২৬
প্রশ্ন
For which real k does x² + 8x + k = 0 have two distinct real roots?
উত্তর
k < 16. D = 64 − 4k must be positive. At k = 16 there is only one distinct real root, so exclude the boundary.
কার্ড ২৭
প্রশ্ন
How many distinct real roots does −2x² + 8x − 8 = 0 have?
উত্তর
One. D = 8² − 4(−2)(−8) = 0. The equation is −2(x − 2)² = 0, so the repeated root is 2.
কার্ড ২৮
প্রশ্ন
For which real k is (k − 2)x² + 3x + 1 = 0 a quadratic equation?
উত্তর
k ≠ 2. The x² coefficient is k − 2, which must be nonzero.
কার্ড ২৯
প্রশ্ন
For which real t does x² + 2tx + 9 = 0 have no real roots?
উত্তর
−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.
কার্ড ৩০
প্রশ্ন
For which real k does x² + 10x + k = 0 have no real roots?
উত্তর
k > 25. D = 100 − 4k must be negative. At k = 25 the repeated root is −5, so exclude that boundary.
কার্ড ৩১
প্রশ্ন
For which real k does x² + kx − 4 = 0 have two distinct real roots?
উত্তর
Every real k. D = k² + 16 is always positive, and the leading coefficient stays 1.
কার্ড ৩২
প্রশ্ন
At k = 2, how many real solutions does (k − 2)x² + 3x + 1 = 0 have?
উত্তর
One: x = −1/3. At k = 2 the x² term vanishes and the equation becomes 3x + 1 = 0; use the linear equation.
কার্ড ৩৩
প্রশ্ন
For which real k does kx² + 2x + 1 = 0 have exactly one distinct real solution?
উত্তর
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.
কার্ড ৩৪
প্রশ্ন
For which real t does x² + 2tx + 9 = 0 have two distinct real roots?
উত্তর
t < −3 or t > 3. D = 4t² − 36 > 0 gives t² > 9. Both boundary values give a repeated root and are excluded.
কার্ড ৩৫
প্রশ্ন
For which real k does kx² + 6x + 2 = 0 have no real solutions?
উত্তর
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.
কার্ড ৩৬
প্রশ্ন
At k = 4, how many real solutions does (k − 4)x² + (k − 4)x + 1 = 0 have?
উত্তর
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.
কার্ড ৩৭
প্রশ্ন
At k = 0, how many real solutions does kx² + kx = 0 have?
উত্তর
Infinitely many: every real x. The equation becomes 0 = 0. Do not apply the quadratic D = 0 rule when the leading coefficient is zero.
কার্ড ৩৮
প্রশ্ন
How many distinct real roots does x² = 0 have?
উত্তর
One: x = 0. D = 0; zero is a repeated root, counted once when counting distinct roots.
কার্ড ৩৯
প্রশ্ন
Do x² − 2x − 2 = 0 and −3x² + 6x + 6 = 0 have different numbers of distinct real roots?
উত্তর
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.
কার্ড ৪০
প্রশ্ন
For which real k does (k − 4)x² + (k − 4)x + 1 = 0 have two distinct real roots?
উত্তর
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.
৪০টি কার্ড
Discriminant Flashcards: Real Roots & Parameter Boundaries
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