Egypt Secondary 1 Mathematics, Week 2: Using the discriminant: finding an unknown k for equal roots and solving with the general formula when the roots are complex

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

Week 2 · The discriminant and the type of the two roots of a quadratic equation, and the sum and product of the two roots · 3 lessons

Using the discriminant: finding an unknown k for equal roots and solving with the general formula when the roots are complex

Mathematics Sec 1 Term 1, Unit 1, Lessons 2–3

GoalFind an unknown coefficient from the kind of roots, and solve an equation with complex roots.

New words

standard form · ax² + bx + c = 0 with everything on one sideno real roots · the discriminant is negative (the roots are complex, not absent)

Explanation

When an equation contains an unknown k, the kind of roots gives a condition on k. Equal roots: b² − 4ac = 0. Two different real roots: b² − 4ac > 0. Complex roots: b² − 4ac < 0. These are three different conditions; read the wording carefully. First collect all terms on one side and read a, b and c (they may contain k). Then write the discriminant, set it equal to 0 or compare it with 0, and solve for k. If the condition is an equation such as (k − 3)² = 16, remember the two solutions k − 3 = 4 and k − 3 = −4. Watch the word: negative discriminant means no REAL roots, but there are still two complex roots, and we can find them with the formula, for example 2x² − 2x + 5 = 0 has b² − 4ac = 4 − 40 = −36, so x = (2 ± 6i)/4 = (1 ± 3i)/2.

Condition on k	What it means
b² − 4ac = 0	equal roots
b² − 4ac > 0	two different real roots
b² − 4ac < 0	complex roots

Worked examples

Find k if x² + 2(k − 3)x + 16 = 0 has equal roots. Here a = 1, b = 2(k − 3), c = 16. b² − 4ac = 4(k − 3)² − 64 = 0, so (k − 3)² = 16 and k − 3 = ±4. So k = 7 or k = −1. Check: k = 7 gives x² + 8x + 16 = (x + 4)², and k = −1 gives x² − 8x + 16 = (x − 4)².

Find k if x² + 6x + k = 2x + 8 has equal roots. Collect: x² + 4x + (k − 8) = 0. Then b² − 4ac = 16 − 4(k − 8) = 48 − 4k = 0, so k = 12. Check: x² + 4x + 4 = (x + 2)² ✓.

For which values of k does kx² − 8x + 4 = 0 have complex roots? b² − 4ac = 64 − 16k < 0, so 16k > 64 and k > 4. (Check k = 5: 64 − 80 = −16 < 0 ✓.)

Class activity

Equal, different or complex? Choose any number for k in x² − 6x + k = 0 and decide the kind of roots. Then test k = 9, k = 5 and k = 12. Which k separates real roots from complex roots?

Practice

Find k if x² + 2kx + 25 = 0 has equal roots.

For which k does x² − 6x + k = 0 have two different real roots?

Solve x² − 4x + 13 = 0 with the general formula.

Show that x² − kx − 3 = 0 has two different real roots for every real k.

A student says: “b² − 4ac < 0 means the equation has no roots.” Correct the sentence.