25
Formulas for the roots of a quadratic equation. The discriminant
Textbook: pp. 141–148
GoalKnow completing the square and the root formula; use the discriminant to find the number of roots and solve quadratic equations.
New words
discriminant · diskriminantcompleting the square · to‘la kvadratni ajratishroot formula · ildizlar formulasireal roots · haqiqiy ildizlar
Explanation
Solving the general equation ax² + bx + c = 0 by completing the square gives the root formula x₁,₂ = (-b ± √(b² - 4ac))/(2a). The expression D = b² - 4ac under the root is called the discriminant of the equation. If D > 0 the equation has two different roots; if D = 0 it has one root x = -b/(2a); if D < 0 it has no real roots. So we compute D first and then use the formula according to its sign. In completing the square we write x² + 6x as (x + 3)² - 9 and reduce the equation to (x + 3)² = k.
Worked examples
2x² - 5x + 2 = 0: D = 25 - 16 = 9 > 0; x = (5 ± 3)/4, x₁ = 2, x₂ = 1/2. x² + 6x + 9 = 0: D = 0, x = -3. x² + 2x + 5 = 0: D = 4 - 20 = -16 < 0, no real root.
Completing the square: x² + 8x + 7 = 0 → x² + 8x + 16 = 9 → (x + 4)² = 9 → x + 4 = ±3 → x₁ = -1, x₂ = -7.
Class activity
“Discriminant lottery”: each pupil picks a, b, c at random, computes D and announces the number of roots; points go for finding more roots.
Practice
1
Find the discriminant of 3x² + 2x - 1 = 0.
16
2
Solve x² - 7x + 10 = 0.
x₁ = 5, x₂ = 2
3
Solve 3x² - 4x + 1 = 0.
x₁ = 1, x₂ = 1/3
4
Why does an equation with D < 0 have no real roots?
Because the formula requires the square root of a negative number, which is impossible among real numbers.