The discriminant b² − 4ac: two different real roots, two equal real roots or two complex conjugate roots
Mathematics Sec 1 Term 1, Unit 1, Lessons 2–3
GoalUse the discriminant to say what kind of roots a quadratic equation has, without solving it.
New words
discriminant · the number b² − 4ac, which decides the kind of rootsrepeated root · a root that counts twice, when both roots are equalconjugate roots · two complex roots p + qi and p − qi
Explanation
Refresher: the roots of ax² + bx + c = 0 (a ≠ 0, with a, b, c real) are x = (−b ± √(b² − 4ac)) / 2a. Everything depends on the number under the square root, called the discriminant, b² − 4ac. If it is positive, √ is a real number and the ± gives two different real roots. If it is zero, ± 0 changes nothing: both roots equal −b/2a, a repeated root. If it is negative, the square root is imaginary, so we get two complex roots (−b ± √(4ac − b²) × i) / 2a, and they are conjugates of each other. On the graph of y = ax² + bx + c the roots are where the curve meets the x-axis: twice, once (it touches) or never. Always write the equation as ax² + bx + c = 0 first, then read a, b and c with their signs.
Worked examples
3x² + 2x − 5 = 0. Here a = 3, b = 2, c = −5, so b² − 4ac = 4 + 60 = 64 > 0: two different real roots. The formula gives x = (−2 ± 8)/6, so x = 1 or x = −5/3.
x² − 6x + 9 = 0. b² − 4ac = 36 − 36 = 0: equal roots. The repeated root is x = −b/2a = 3. Indeed x² − 6x + 9 = (x − 3)². The curve touches the x-axis at (3, 0).
2x² − 4x + 5 = 0. b² − 4ac = 16 − 40 = −24 < 0: no real root, two complex roots. √(−24) = 2√6 i, so x = (4 ± 2√6 i)/4 = 1 ± (√6/2) i.
Class activity
Three curves. Look at the graph above. For each of the three parabolas, count where it meets the x-axis and match it with the sign of its discriminant.
Practice
1
Find the discriminant and the kind of roots of 2x² − 7x + 3 = 0.
49 − 24 = 25 > 0: two different real roots (x = 3 and x = 1/2).
2
What kind of roots has x² + 4x + 4 = 0?
16 − 16 = 0: two equal real roots, x = −2 twice.
3
What kind of roots has x² + 2x + 7 = 0?
4 − 28 = −24 < 0: two complex conjugate roots.
4
Write x(x − 3) = 10 in standard form and find the kind of roots.
x² − 3x − 10 = 0; 9 + 40 = 49 > 0: two different real roots (5 and −2).
5
What kind of roots has 3x² = 12x − 12?
3x² − 12x + 12 = 0; 144 − 144 = 0: two equal real roots, x = 2.
