Find the zeroes of the quadratic polynomial 5x² − 8x − 4 and verify the relationship between the zeroes and the coefficients.

1. 1 · Write what is given and what is to be found. p(x) = 5x² − 8x − 4, so a = 5, b = −8, c = −4. To find: the zeroes, then verify.
2. 2 · Write the formula. Sum of zeroes = −b/a. Product of zeroes = c/a.
3. 3 · Factorise by splitting the middle term. ac = −20. Two numbers with product −20 and sum −8 are −10 and 2. 5x² − 10x + 2x − 4 = 5x(x − 2) + 2(x − 2) = (5x + 2)(x − 2).
4. 4 · Find the zeroes. x = 2 and x = −2/5.
5. 5 · Verify the sum. 2 + (−2/5) = 8/5, and −b/a = −(−8)/5 = 8/5.
6. 6 · Verify the product. 2 × (−2/5) = −4/5, and c/a = −4/5.
7. 7 · Check and box the answer. The zeroes are 2 and −2/5. Therefore the relationship is verified.

Scoring tip: Write a, b and c with their signs before you start. Most lost marks come from the sign of b.