Equations and inequalities

Higher-degree equations

Equations with x³ or x⁴ look hard but almost always reduce to a quadratic. Entrance tests use three standard tricks.

München Named for the T-Kurs maths test at Studienkolleg München. Source

The rules

• Factor out x: x³ − 4x = x(x² − 4) = 0.
• Biquadratic equations: substitute z = x², solve the quadratic in z, then take ±√z.
• Guess a whole-number root (try divisors of the constant term), then divide by (x − root).
• A product is zero when one factor is zero.

Practice questions with solutions

Generated by the same engine as the Kollego course and solvable without a calculator. Pick an answer in your head, then open the solution.

1. 1. Solve x³ = 8. (real solution)

   • a2
   • b4
   • c8
   • d-2
   Show solution

   Answer: 2

   Take the cube root of both sides. x³ has exactly one real cube root.

   x = ∛8 = 2.

2. 2. Solve x³ = 125. (real solution)

   • a25
   • b-5
   • c125
   • d5
   Show solution

   Answer: 5

   Take the cube root of both sides. x³ has exactly one real cube root.

   x = ∛125 = 5.

3. 3. Solve x³ = 27. (real solution)

   • a3
   • b9
   • c27
   • d-3
   Show solution

   Answer: 3

   Take the cube root of both sides. x³ has exactly one real cube root.

   x = ∛27 = 3.

4. 4. Solve x³ = 64. (real solution)

   • a4
   • b64
   • c16
   • d-4
   Show solution

   Answer: 4

   Take the cube root of both sides. x³ has exactly one real cube root.

   x = ∛64 = 4.

Common mistakes

• Forgetting the negative root when going back from z = x².
• Keeping negative values of z, which give no real x.
• Dividing by x and losing x = 0.

Related topics

All maths topics · T-Kurs · M-Kurs · W-Kurs