Equations with transformed roots

Method

1. Find α + β and αβ from the original equation.

2. Find the sum of the new roots and the product of the new roots, in terms of α + β and αβ.

3. Write x² - (new sum)x + (new product) = 0, then multiply to give integer coefficients.

Common transformations

• Roots α + h, β + h: sum (α + β) + 2h; product αβ + h(α + β) + h².
• Roots kα, kβ: sum k(α + β); product k² αβ.
• Roots 1/α, 1/β: sum (α + β)/(αβ); product 1/(αβ).
• Roots α², β²: sum (α + β)² - 2αβ; product (αβ)².
• Roots α³, β³: sum (α + β)³ - 3αβ(α + β); product (αβ)³.

Presentation

The answer is an equation: include "= 0", use integer coefficients, and do not drop the x term. Check one easy case if you can, for example by solving the original equation mentally when its roots are simple.

Worked example

α and β are the roots of 2x² - 3x - 4 = 0. Find a quadratic equation with integer coefficients whose roots are α² and β².

1. α + β = 3/2 and αβ = -2.
2. New sum = (α + β)² - 2αβ = 9/4 + 4 = 25/4. New product = (αβ)² = 4.
3. x² - (25/4)x + 4 = 0; multiply by 4.

Answer: 4x² - 25x + 16 = 0

Check yourself

1. α and β are the roots of 2x² - 3x - 4 = 0. Find a quadratic with integer coefficients and roots α + 3 and β + 3. Give the left-hand side of the equation, with x² coefficient 2.
   Show answer

   2x² - 15x + 23. Sum = 3/2 + 6 = 15/2. Product = αβ + 3(α + β) + 9 = -2 + 9/2 + 9 = 23/2. x² - (15/2)x + 23/2 = 0, so 2x² - 15x + 23 = 0.

2. Using the same α and β, give the left-hand side of the integer quadratic equation with roots α² and β², with x² coefficient 4.
   Show answer

   4x² - 25x + 16. Sum 25/4 and product 4 give x² - (25/4)x + 4 = 0.

3. Using the same α and β, give the left-hand side of the integer quadratic equation with roots 1/α and 1/β, with x² coefficient 4.
   Show answer

   4x² + 3x - 2. Sum = (α + β)/(αβ) = -3/4. Product = 1/(αβ) = -1/2. x² + (3/4)x - 1/2 = 0, so 4x² + 3x - 2 = 0.

4. Using the same α and β, give the left-hand side of the integer quadratic equation with roots α³ and β³, with x² coefficient 8.
   Show answer

   8x² - 99x - 64. Sum = 99/8 (from the worked identity) and product = (αβ)³ = -8. x² - (99/8)x - 8 = 0, so 8x² - 99x - 64 = 0.

5. Using the same α and β, give the left-hand side of the quadratic equation with roots 2α and 2β, with x² coefficient 1.
   Show answer

   x² - 3x - 8. Sum = 2(3/2) = 3. Product = 4αβ = -8. So x² - 3x - 8 = 0.

6. The roots of a quadratic are α and β. What is the product of the new roots α + 1 and β + 1?
   • αβ + 1
   • αβ + 2
   • αβ + (α + β) + 1
   • (α + β) + 2
   Show answer

   αβ + (α + β) + 1. (α + 1)(β + 1) = αβ + α + β + 1.

7. A student's final answer is x² - (25/4)x + 4. What should be improved?
   • Multiply by 4 to get integer coefficients and write = 0
   • Nothing: it is fully correct
   • Change the sign of the x term
   • Divide by 4 so the constant is 1
   Show answer

   Multiply by 4 to get integer coefficients and write = 0. The question asks for an equation with integer coefficients: 4x² - 25x + 16 = 0.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics, Further Mathematics and Pure Mathematics specification (issue 3), Unit FP1
• Pearson Edexcel IAL Further Pure Mathematics 1 scheme of work

Last updated 8 November 2026. Spotted a mistake? Tell us.