The Discriminant & Nature of Roots: Full Guide — Worksheet & Memo

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Grade 11 · Equations

The Discriminant & Nature of Roots: Full Guide

MARKING GUIDELINE

Marks

Duration

50 minutes

Questions

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Class:

Date:

Mark

/ 33

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
The DISCRIMINANT of ax² + bx + c = 0 is …
(1)
A)b² − 4ac✓
B)b² + 4ac
C)−b² − 4ac
D)√(b² − 4ac)
Answer: A — Δ = b² − 4ac — the expression under the root in the quadratic formula.
- B — the sign is a MINUS, not a plus
- C — the −b belongs outside the root, not inside it
- D — the discriminant is that expression itself, not its square root
1.2
If Δ > 0 AND Δ is a perfect square, the roots are …
(1)
A)real, rational and unequal✓
B)real, irrational and unequal
C)real and equal
D)non-real
Answer: A — A perfect-square discriminant has a whole-number square root, so the roots come out rational.
- B — irrational roots come from a discriminant that is positive but NOT a perfect square
- C — equal roots need Δ = 0
- D — non-real roots need Δ < 0
1.3
If Δ = 0, the roots are …
(1)
A)real and equal✓
B)real and unequal
C)non-real
D)one real and one non-real
Answer: A — With Δ = 0 the ±√Δ term vanishes, leaving one repeated root.
- B — unequal roots need Δ > 0
- C — non-real roots need Δ < 0
- D — the roots of a real quadratic are either both real or both non-real
1.4
For x² − 6x + 9 = 0, the roots are …
(1)
A)real and equal✓
B)real and unequal
C)non-real
D)rational and unequal
Answer: A — Δ = (−6)² − 4(1)(9) = 36 − 36 = 0.
- B — Δ = 0, which is not greater than 0
- C — Δ = 0, which is not negative
- D — Δ = 0 gives ONE repeated root, so the roots are equal
1.5
For 2x² + 3x + 5 = 0, the roots are …
(1)
A)non-real✓
B)real and equal
C)real and rational
D)real and irrational
Answer: A — Δ = 3² − 4(2)(5) = 9 − 40 = −31, which is negative.
- B — equal roots need Δ = 0, not Δ < 0
- C — a negative discriminant gives no real roots at all
- D — a negative discriminant gives no real roots, irrational ones included
1.6
For x² − 5x + 3 = 0, the roots are …
(1)
A)real, irrational and unequal✓
B)real, rational and unequal
C)real and equal
D)non-real
Answer: A — Δ = 25 − 12 = 13, which is positive but not a perfect square.
- B — 13 is not a perfect square, so the roots are irrational
- C — equal roots need Δ = 0
- D — Δ = 13 is positive, so the roots ARE real
1.7
For which value(s) of k does x² + kx + 4 = 0 have EQUAL roots?
(1)
A)k = 4 or k = −4✓
B)k = 4 only
C)k = 16
D)k = 2
Answer: A — Δ = k² − 16 = 0 gives k² = 16, so k = ±4.
- B — k² = 16 has TWO solutions
- C — forgot to take the square root of 16
- D — square-rooted twice instead of once
1.8
If Δ < 0, the graph of the quadratic …
(1)
A)does not cut the x-axis✓
B)touches the x-axis once
C)cuts the x-axis twice
D)lies entirely below the x-axis
Answer: A — No real roots means no x-intercepts at all.
- B — that happens when Δ = 0
- C — that happens when Δ > 0
- D — it lies wholly on ONE side of the axis, but which side depends on the sign of a
1.9
For 3x² − 12x + 12 = 0, Δ = …
(1)
A)0✓
B)144
C)288
D)−144
Answer: A — Δ = (−12)² − 4(3)(12) = 144 − 144 = 0.
- B — worked out b² but never subtracted 4ac
- C — ADDED 4ac instead of subtracting it
- D — subtracted b² from 4ac instead of the other way round
1.10
For which values of k does x² + 2x + k = 0 have REAL roots?
(1)
A)k ≤ 1✓
B)k ≥ 1
C)k < 1
D)k = 1
Answer: A — Δ = 4 − 4k ≥ 0 gives 4 ≥ 4k, so k ≤ 1.
- B — did not reverse the inequality when moving the k across
- C — k = 1 gives Δ = 0, and equal roots ARE real, so 1 is included
- D — that gives equal roots only, not every case of real roots

Answer grid — marking guideline

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[9 MARKS]

Use the discriminant Δ = b² − 4ac to answer the questions below.

2.1
Describe the nature of the roots of x² − 4x + 4 = 0.
(3)
Δ = 16 − 16 (2) = 0 (1)
The roots are real, rational and EQUAL
2.2
Describe the nature of the roots of x² + 3x + 5 = 0.
(3)
Δ = 9 − 20 (2) = −11 (1)
Δ < 0, so the roots are NON-REAL
2.3
Describe the nature of the roots of 2x² − 7x + 3 = 0.
(3)
Δ = 49 − 24 = 25 (2)
Δ > 0 and a perfect square, so the roots are real, rational and unequal (1)

Question 3

[14 MARKS]

Answer the questions below.

3.1
Determine the value(s) of k for which kx² + 4x + 2 = 0 has REAL roots.
(5)
Real roots means Δ ≥ 0 (1)
16 − 8k ≥ 0 (2)
k ≤ 2 (1)
k ≠ 0, or the equation is not quadratic: k ≤ 2, k ≠ 0 (1)
3.2
Determine the value(s) of p for which x² + px + 9 = 0 has equal roots.
(4)
p² − 36 = 0 (2)
p² = 36 (1)
p = 6 or p = −6 (1)
3.3
State what the discriminant tells you about the GRAPH of y = ax² + bx + c.
(3)
Δ > 0: it cuts the x-axis twice (1)
Δ = 0: it touches once (1)
Δ < 0: it never meets the x-axis (1)
3.4
Explain the difference between roots that are RATIONAL and roots that are IRRATIONAL.
(2)
Rational roots occur when Δ is a perfect square (1); otherwise the surd does not simplify and the roots are irrational (1)

TOTAL: 33 marks

This question paper consists of 3 questions.