Solving disguised quadratics

In some cases, we might see a polynomial that has a higher degree than (the degree of a quadratic), but it may still be possible to solve it by treating it like a quadratic equation. These are called disguised quadratics.

What is a disguised quadratic?

A disguised quadratic is a polynomial where we can substitute a variable to transform it into a quadratic form. This usually involves us spotting a pattern in the powers in the expression.

Disguised quadratics where powers are multiples

A common type of disguised quadratic is one where the powers are consecutive multiples of . For example, one in the form . In this case, we can substitute , which transforms the equation into , which is a normal quadratic in terms of .

Example: Solve the equation

• Substitute :
• Factorise the quadratic:
• Find the values of :
  • or
• Substitute back to find :
  • If , then :
  • If , then :

Answer: or

Disguised quadratics where terms are exponents

Solve the equation

• Substitute :
• Factorise the quadratic:
• Find the values of :
  • or
• Substitute back to find :
  • If , then :
  • If , then :
    • Answer: or

Solve the equation

• Substitute :
• Factorise the quadratic:
• Find the values of :
  • or
• Substitute back to find :
  • If , then :
    • No solution, since is always positive.
  • If , then :
    • Answer: