Algebra & Identities • Topic 4 of 4

Quadratic Equations

A quadratic ax^2 + bx + c = 0 is solved by factorising, completing the square, or the formula x = (-b +/- root(b^2-4ac))/2a. The discriminant b^2 - 4ac tells the nature of roots (positive: two real, zero: equal, negative: imaginary). By Vieta's relations, sum of roots = -b/a and product = c/a — handy for forming an equation from its roots.

✅ Solved examples

1. Solve x^2 - 5x + 6 = 0.
Factor (x-2)(x-3) = 0 -> x = 2 or 3.
2. Sum and product of roots of x^2 - 7x + 10 = 0?
Sum = 7, product = 10 (roots 2 and 5).
3. Form a quadratic whose roots are 3 and -4.
Sum -1, product -12: x^2 + x - 12 = 0.
4. Nature of roots of x^2 + 4x + 5 = 0?
Discriminant 16 - 20 = -4 < 0 -> imaginary (no real roots).

✏️ Practice — try these, take hints as needed

1. Solve x^2 - 9x + 20 = 0.
Factor.
(x-4)(x-5).
—
x = 4 or 5
2. Sum of roots of x^2 - 6x + 8 = 0?
-b/a.
6.
—
6
3. Product of roots of 2x^2 - 3x - 5 = 0?
c/a.
-5/2.
—
-2.5
4. Quadratic with roots 2 and 7?
Sum 9, product 14.
x^2 - 9x + 14.
—
x^2 - 9x + 14 = 0
5. Discriminant of x^2 - 4x + 4 = 0?
16 - 16.
0 -> equal roots.
—
0 (equal roots)

📝 Topic test — 8 questions

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