Discriminant Flashcards: Real Roots & Parameter Boundaries

Question 1

For real coefficients in ax² + bx + c = 0, with a ≠ 0, what is the discriminant?

Accepted Answer

D = b² − 4ac. Its sign tells you the number of distinct real roots.

Question 2

Rewrite x² = 5x + 1 in zero form before using the discriminant.

Accepted Answer

x² − 5x − 1 = 0. Use a = 1, b = −5 and c = −1.

Question 3

In −2x² + 7x − 4 = 0, what are a, b and c?

Accepted Answer

a = −2, b = 7, c = −4. Keep the signs attached to the coefficients.

Question 4

Before using the quadratic discriminant on ax² + bx + c = 0, which coefficient must be nonzero?

Accepted Answer

a ≠ 0. If a = 0, solve bx + c = 0 separately; the equation is no longer quadratic.

Question 5

A real quadratic has D > 0. How many distinct real roots?

Accepted Answer

Two distinct real roots. The two square-root choices in the quadratic formula give different real values.

Question 6

Find the discriminant of 3x² − 2x − 4 = 0.

Accepted Answer

52

D = (−2)² − 4(3)(−4) = 4 + 48 = 52.

Question 7

In 4x² + 9 = 0, what value of b belongs in D = b² − 4ac?

Accepted Answer

b = 0. There is no x term; c = 9 is the constant coefficient.

Question 8

A real quadratic has a negative x² coefficient. Does that alone determine its number of real roots?

Accepted Answer

No. The parabola opens downward, but D can be positive, zero or negative.

Question 9

A real quadratic has D = 0. How many distinct real roots?

Accepted Answer

One distinct real root, repeated twice. Its value is −b/(2a).

Question 10

How many distinct real roots does 6x² + 5x − 6 = 0 have?

Accepted Answer

Two. D = 5² − 4(6)(−6) = 169 > 0. You do not need to solve for the roots to count them.

Question 11

For b = −5, a student uses b² = −25 in the discriminant. What should replace −25?

Accepted Answer

25

Square the whole signed coefficient: (−5)² = 25.

Question 12

For x² + 4x = 2, a student uses c = 2. What is the correct discriminant?

Accepted Answer

24

First write x² + 4x − 2 = 0, so c = −2 and D = 4² − 4(1)(−2) = 24.

Question 13

A real quadratic has D < 0. How many real roots?

Accepted Answer

None. A negative discriminant has no real square root.

Question 14

How many distinct real roots does 4x² + 12x + 9 = 0 have?

Accepted Answer

One. D = 12² − 4(4)(9) = 0. The equation is (2x + 3)² = 0, so its repeated root is −3/2.

Question 15

How many distinct real roots does x² − 7x = 0 have?

Accepted Answer

Two. D = (−7)² − 4(1)(0) = 49 > 0. Factoring gives x(x − 7) = 0, with roots 0 and 7.

Question 16

What discriminant condition is necessary and sufficient for a real quadratic to have two distinct real roots?

Accepted Answer

D > 0. The condition assumes the x² coefficient is nonzero.

Question 17

How many real roots does 2x² + 4x + 5 = 0 have?

Accepted Answer

None. D = 4² − 4(2)(5) = −24 < 0.

Question 18

A real quadratic graph touches the x-axis at exactly one point. What is its discriminant?

Accepted Answer

D = 0. The touching point corresponds to a repeated real root.

Question 19

How many distinct real roots does −3x² + 2x + 2 = 0 have?

Accepted Answer

Two. D = 2² − 4(−3)(2) = 28 > 0. A negative leading coefficient does not prevent real roots.

Question 20

For x² + 5 = 0, a student uses b = 5 and c = 0. What is the correct discriminant?

Accepted Answer

−20. The coefficients are a = 1, b = 0 and c = 5, so D = 0² − 4(1)(5) = −20.

Question 21

What discriminant condition is necessary and sufficient for a real quadratic to have no real roots?

Accepted Answer

D < 0. The coefficients are real and the x² coefficient is nonzero.

Question 22

For which real k does x² − 6x + k = 0 have a repeated root?

Accepted Answer

k = 9. D = 36 − 4k = 0. At that value the equation is (x − 3)² = 0.

Question 23

Does D = 0 mean the root is zero?

Accepted Answer

No. It means the two roots coincide. For (3x − 2)² = 0, the repeated root is 2/3, while D = 0.

Question 24

A real quadratic equation is multiplied throughout by a nonzero real λ. How does its discriminant change?

Accepted Answer

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.

Question 25

An upward-opening real quadratic graph has its vertex strictly below the x-axis. What is the sign of D?

Accepted Answer

Positive: D > 0. The graph crosses the x-axis twice. The vertex must be strictly below, not on, the axis.

Question 26

For which real k does x² + 8x + k = 0 have two distinct real roots?

Accepted Answer

k < 16. D = 64 − 4k must be positive. At k = 16 there is only one distinct real root, so exclude the boundary.

Question 27

How many distinct real roots does −2x² + 8x − 8 = 0 have?

Accepted Answer

One. D = 8² − 4(−2)(−8) = 0. The equation is −2(x − 2)² = 0, so the repeated root is 2.

Question 28

For which real k is (k − 2)x² + 3x + 1 = 0 a quadratic equation?

Accepted Answer

k ≠ 2. The x² coefficient is k − 2, which must be nonzero.

Question 29

For which real t does x² + 2tx + 9 = 0 have no real roots?

Accepted Answer

−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.

Question 30

For which real k does x² + 10x + k = 0 have no real roots?

Accepted Answer

k > 25. D = 100 − 4k must be negative. At k = 25 the repeated root is −5, so exclude that boundary.

Question 31

For which real k does x² + kx − 4 = 0 have two distinct real roots?

Accepted Answer

Every real k. D = k² + 16 is always positive, and the leading coefficient stays 1.

Question 32

At k = 2, how many real solutions does (k − 2)x² + 3x + 1 = 0 have?

Accepted Answer

One: x = −1/3. At k = 2 the x² term vanishes and the equation becomes 3x + 1 = 0; use the linear equation.

Question 33

For which real k does kx² + 2x + 1 = 0 have exactly one distinct real solution?

Accepted Answer

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.

Question 34

For which real t does x² + 2tx + 9 = 0 have two distinct real roots?

Accepted Answer

t < −3 or t > 3. D = 4t² − 36 > 0 gives t² > 9. Both boundary values give a repeated root and are excluded.

Question 35

For which real k does kx² + 6x + 2 = 0 have no real solutions?

Accepted Answer

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.

Question 36

At k = 4, how many real solutions does (k − 4)x² + (k − 4)x + 1 = 0 have?

Accepted Answer

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.

Question 37

At k = 0, how many real solutions does kx² + kx = 0 have?

Accepted Answer

Infinitely many: every real x. The equation becomes 0 = 0. Do not apply the quadratic D = 0 rule when the leading coefficient is zero.

Question 38

How many distinct real roots does x² = 0 have?

Accepted Answer

One: x = 0. D = 0; zero is a repeated root, counted once when counting distinct roots.

Question 39

Do x² − 2x − 2 = 0 and −3x² + 6x + 6 = 0 have different numbers of distinct real roots?

Accepted Answer

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.

Question 40

For which real k does (k − 4)x² + (k − 4)x + 1 = 0 have two distinct real roots?

Accepted Answer

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.

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Discriminant Flashcards: Real Roots & Parameter Boundaries