A cubic discriminates real from complex roots

(i) One root of x3−3x+c=0x^3-3x+c=0, where cc is real, is 2−3i2-3i. Without using a calculator, find cc and the other roots. marks

(ii) If cc is a nonzero purely imaginary number, explain whether x3−3x+c=0x^3-3x+c=0 can have real roots. mark

(iii) Instead let cc be real. By considering y=x3−3xy=x^3-3x, find the range of cc for which x3−3x+c=0x^3-3x+c=0 has only real roots. marks

[2020/VJC/Prelim/I/5]

Total marks: 8

Solution

(i)

Conjugate roots and Vieta

One root of x3−3x+c=0x^3-3x+c=0, with cc real, is 2−3i2-3i. Without a calculator, find cc and the other roots.

Explore this idea

Hint

  1. State the conjugate root.

  2. Use that the roots sum to 00 to find the third root, then use the product to find cc.

Answer · (i)

c=52c=52; other roots 2+3i,−42+3i,-4.

Explore the idea

The roots as c changes

Slide cc and watch the three roots. At c=52c=52 you see 2±3i2\pm3i and −4-4.

Value of c52
Try an example
root 1
-4
root 2
2 + 3i
root 3
2 − 3i
  • Moves as the slider changes; a faint path shows where it goes
Key idea

A complex pair 2±3i2\pm3i forces the real root to be −4-4 because the three roots must sum to zero.

Ideas to try
  1. Set c=52c=52 and check that the three roots add to 00: 4+(−4)=04+(-4)=0.
  2. Reduce cc toward 00. The roots meet the real axis in a different way each time.
(ii)

Explain why there are no real roots

If cc is a nonzero purely imaginary number, can x3−3x+c=0x^3-3x+c=0 have real roots?

Explore this idea

Hint

  1. For real xx, what kind of number is x3−3xx^3-3x?

Answer · (ii)

No: real x3−3xx^3-3x cannot cancel nonzero purely imaginary cc.

Explore the idea

No root ever touches the real axis

Let c=sic=si and slide ss. None of the three roots lies on the horizontal axis.

Value of s in c = s i2
Try an example
root 1
-1.807 − 0.298i
root 2
0.596i
root 3
1.807 − 0.298i
  • Moves as the slider changes; a faint path shows where it goes
Key idea

With an imaginary cc the roots are never conjugate pairs, and none of them is real.

Ideas to try
Look at the imaginary parts in the readouts: none of them is 00.
(iii)

Count roots from a graph

For real cc, by considering y=x3−3xy=x^3-3x, find the range of cc for which x3−3x+c=0x^3-3x+c=0 has only real roots.

Explore this idea

Hint

  1. Rewrite as x3−3x=−cx^3-3x=-c.

  2. Find the local maximum and minimum of the cubic.

Answer · (iii)

−2≤c≤2-2\le c\le2.

Repeated roots allowed; three distinct real roots require −2<c<2.

Explore the idea

The roots leave the real axis at c = ±2

Slide cc between −4-4 and 44. For ∣c∣≤2|c|\le2 every root is real; beyond that, a pair moves off the axis.

Value of c0
Try an example
root 1
-1.732
root 2
0
root 3
1.732
  • Moves as the slider changes; a faint path shows where it goes
Key idea

Roots can only become non-real in pairs, at the moment two real roots collide. That happens at the turning points of the curve.

Ideas to try
  1. At c=2c=2 two roots coincide at x=1x=1. Push cc past 22 and they split into a conjugate pair.
  2. At c=−2c=-2 the double root is at x=−1x=-1, and a pair appears when c<−2c<-2.
Takeaway

Roots collide before they leave the line

A pair of real roots can become complex only through a repeated root. The turning points of the graph tell you where.