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Quadratic equations: the discriminant and the sum and product of the roots

الأسبوع 2 · المميز ونوع جذري المعادلة التربيعية، ومجموع جذري المعادلة وحاصل ضربهما · 3 دروس
3

The sum and the product of the two roots, −b/a and c/a, found without solving the equation

Mathematics Sec 1 Term 1, Unit 1, Lessons 2–3
GoalFind the sum and the product of the roots from the coefficients, and use them to find unknown coefficients.
New words
sum of the roots · L + M = −b/aproduct of the roots · L × M = c/a
Explanation

Call the roots L and M. From the formula L = (−b + √D)/2a and M = (−b − √D)/2a, where D = b² − 4ac. Adding them, the square roots cancel and L + M = −2b/2a = −b/a. Multiplying them, L × M = (b² − D)/4a² = 4ac/4a² = c/a. These two facts hold for every quadratic, even when the roots are complex, and we never have to solve the equation. Check with 3x² − 7x − 6 = 0, which factorises as (3x + 2)(x − 3): the roots are −2/3 and 3. Their sum is 7/3 = −(−7)/3 and their product is −2 = −6/3 ✓. Remember: write the equation in standard form first, keep the minus sign in −b/a, and note that the product is c/a, not −c/a.

ƒFormula
L + M = −b/a L × M = c/a
for ax² + bx + c = 0, a ≠ 0
Worked examples
Find the sum and product of the roots of 4x² = 9 − 6x without solving. Standard form: 4x² + 6x − 9 = 0, so a = 4, b = 6, c = −9. Sum = −6/4 = −3/2. Product = −9/4.
The sum of the roots of 3x² + kx − 5 = 0 is 4. Find k. Sum = −k/3 = 4, so k = −12. The equation is 3x² − 12x − 5 = 0.
Find the sum and product of the roots of (2x − 3)(x + 2) = 0. Expand: 2x² + x − 6 = 0. Sum = −1/2, product = −6/2 = −3. Check with the roots 3/2 and −2: 3/2 − 2 = −1/2 ✓ and 3/2 × (−2) = −3 ✓.
Class activity

Check it yourself. Solve x² − 9x + 20 = 0 by factorising, add and multiply the two roots, and compare with −b/a and c/a.

Practice
1
Find the sum and the product of the roots of x² − 9x + 20 = 0.
2
Find the sum and the product of the roots of 5x² + 10x − 3 = 0.
3
Find the sum and the product of the roots of 3x² = 2x + 8.
4
The sum of the roots of 2x² + kx + 6 = 0 is 7/2. Find k and the roots.
5
The product of the roots of 4x² − 12x + c = 0 is 2. Find c and the roots.