A quadratic equation has degree 2, and computing just one number, the discriminant D = b² − 4ac, tells you how many real roots it has before you find them.
If D is positive there are two distinct real roots, if it is 0 a repeated root, and if it is negative two distinct complex roots; this matches exactly whether the parabola y = ax² + bx + c crosses the x-axis twice, touches it, or misses it.
The relationship between roots and coefficients gives the sum and product of the two roots from the coefficients alone, without solving.
In the first picture the way the parabola meets the x-axis changes with b and c; in the second, the two roots and their sum and product change with p and q.
The discriminant tells you how many roots
For the quadratic equation ax² + bx + c = 0, the discriminant D = b² − 4ac tells you, from its sign alone, how many real roots there are. Seeing it alongside the way the parabola meets the x-axis makes the reason clear.
-2
-3
seegongsik.com
Depending on the sign of the discriminant D, the parabola y = x² + bx + c crosses the x-axis twice, touches it or misses it
💡 What the discriminant says
①If D > 0 the parabola meets the x-axis at two points, and the equation has two distinct real roots
②If D = 0 the parabola touches the x-axis, and the equation has a repeated root
③If D < 0 the parabola does not meet the x-axis, and the equation has two distinct complex roots
④In the picture a is fixed at 1, so D = b² − 4c
The quadratic formula
Quadratic formula
x = -b ± √(b² - 4ac)2a
The roots of ax² + bx + c = 0 (a ≠ 0)
Discriminant
D = b² − 4ac
D > 0: two distinct real roots, D = 0: a repeated root, D < 0: two distinct complex roots
📐 Where the formula comes from
①Divide by a and complete the square: (x + b/(2a))² = (b² − 4ac)/(4a²)
②Take square roots of both sides: x + b/(2a) = ±√D/(2a)
③Rearrange for x and you have the quadratic formula
④Because √D appears, the sign of D decides what kind of roots there are
Roots and coefficients
Even without finding the two roots α and β, their sum and product can be read off from the coefficients.
-5
6
seegongsik.com
The two roots α and β of x² + px + q = 0 shown on a number line, with their sum α + β and product αβ
Formulas linking roots and coefficients
Roots and coefficients
α + β = -ba, αβ = ca
Sum and product of the roots α, β of ax² + bx + c = 0
Building a quadratic from its roots
x² − (α + β)x + αβ = 0
Knowing the sum and product, you can write the quadratic with leading coefficient 1 at once
④In the second picture, even when D < 0 and no roots appear on the number line, the sum and product are still worked out, because complex roots satisfy these relationships too
Worked examples and unit test
Example 1
Find the two roots of x² − 5x + 6 = 0.
1
Factorise the left-hand side.
x² − 5x + 6 = (x − 2)(x − 3) = 0
2
Set each factor equal to 0.
x = 2 or x = 3
▸ x = 2 or x = 3
The sum of the roots, 5, and their product, 6, equal −(−5)/1 and 6/1, so the relationship between roots and coefficients checks out.
Example 2
The roots of x² − 7x + 10 = 0 are α and β. Find α² + β².
1
Find the sum and product from the coefficients.
α + β = 7, αβ = 10
2
Substitute into α² + β² = (α + β)² − 2αβ.
= 7² − 2·10 = 49 − 20 = 29
▸ α² + β² = 29
The roots are in fact 2 and 5, and 2² + 5² = 29. You reach the same answer without solving.
Summary
Quadratic formula
x = -b ± √(b² - 4ac)2a
A method that works for every quadratic equation
Roots and coefficients
α + β = -ba, αβ = ca
Sum and product without finding the roots
Grade-10 education-office assessment type
If x² − 4x + k = 0 has a double root, what is the constant k?
①1
②2
③3
④4
⑤5
▸ ④ 4
1
A double root means the discriminant D = 0.
D = b² − 4ac = (−4)² − 4·1·k
2
Solve D = 0 for k.
16 − 4k = 0 ⇒ k = 4
🎯 Points that often come up
①First decide the kind of roots from the discriminant D
②The condition D = 0 is used often in questions on tangents and repeated roots
③Use the relationship between roots and coefficients to find symmetric expressions such as α² + β²
④Build a new quadratic equation from the sum and product of two roots
⑤For a quadratic with real coefficients, complex roots come as a conjugate pair, so knowing one fixes the other