All subjects · Further Mathematics · WFM01 Further Pure Mathematics 1

Sum and product of roots

Edexcel IAL Further Mathematics · WFM01 Further Pure Mathematics 1

You can find the sum and the product of the roots of a quadratic straight from its coefficients, without solving it. This works for real and complex roots alike.

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The results

If α and β are the roots of ax2 + bx + c = 0, then ax2 + bx + c = a(x - α)(x - β) = a(x2 - (α + β)x + αβ).

Comparing coefficients: α + β = -b/a and αβ = c/a. Learn these.

Going backwards

A quadratic with roots α and β is x2 - (α + β)x + αβ = 0. Multiply through to clear any fractions: sum 4/3, product 5/3 gives x2 - (4/3)x + 5/3 = 0, so 3x2 - 4x + 5 = 0. Always write "= 0".

Using conditions on the roots

If one root is twice the other, call them α and 2α. Then 3α = -b/a and 2α2 = c/a: solve for α, then the unknown coefficient.

Worked example

The roots of x2 - 9x + k = 0 are α and 2α. Find k.

  1. Sum: α + 2α = 9, so α = 3.
  2. Product: α × 2α = 2α2 = 18.
  3. Product = c/a = k.

Answer: k = 18

Key idea: α + β = -b/a and αβ = c/a. A quadratic with given roots is x2 - (sum)x + (product) = 0.

Check yourself

  1. α and β are the roots of 2x2 - 5x + 3 = 0. Find α + β. Give your answer as a fraction in its simplest form.
    Show answer

    5/2. α + β = -b/a = -(-5)/2 = 5/2.

  2. α and β are the roots of 2x2 - 5x + 3 = 0. Find αβ. Give your answer as a fraction in its simplest form.
    Show answer

    3/2. αβ = c/a = 3/2.

  3. α and β are the roots of 3x2 + 7x - 2 = 0. Which is correct?
    • α + β = -7/3, αβ = -2/3
    • α + β = 7/3, αβ = -2/3
    • α + β = -7/3, αβ = 2/3
    • α + β = -2/3, αβ = -7/3
    Show answer

    α + β = -7/3, αβ = -2/3. α + β = -b/a = -7/3 and αβ = c/a = -2/3.

  4. The roots of x2 - 9x + k = 0 are α and 2α. Find k.
    Show answer

    18. 3α = 9 so α = 3, and k = 2α2 = 18.

  5. Find a quadratic equation with integer coefficients whose roots have sum 4/3 and product 5/3. Give the left-hand side, with x2 coefficient 3.
    Show answer

    3x2 - 4x + 5. x2 - (4/3)x + 5/3 = 0. Multiply by 3: 3x2 - 4x + 5 = 0.

  6. The roots of z2 + 6z + 25 = 0 are complex. Find their product.
    Show answer

    25. αβ = c/a = 25. Check: the roots are -3 ± 4i and (-3 + 4i)(-3 - 4i) = 9 + 16 = 25.

  7. One root of x2 + px + 12 = 0 is 3 times the other. Given p < 0, find p.
    Show answer

    -8. Let the roots be α and 3α. 3α2 = 12 gives α = ±2; the sum 4α = -p. With p < 0, α = 2, the sum is 8 and p = -8.

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