Part 30 of 30 · QuadraticsSee the whole course
The quadratic formula and the discriminant
For ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a. The number under the root, b² − 4ac, is the discriminant. If it is positive there are two solutions, if it is zero there is one, and if it is negative there are none (no real ones).
When to use it
Factoring is quick when the numbers are friendly. The formula works on every quadratic, including the ones that do not factor into whole numbers. First make sure the equation equals 0, then read off a, b and c with their signs.
The method in three steps
- Write a, b and c, with signs. In x² − 2x − 3 = 0: a = 1, b = −2, c = −3.
- Work out the discriminant b² − 4ac.
- Put everything into the formula: x = (−b ± √D) ÷ 2a. The ± gives two answers, one with plus and one with minus.
Example 1: x² − 2x − 3 = 0
a = 1, b = −2, c = −3.
b² − 4ac = (−2)² − 4(1)(−3) = 4 + 12 = 16.
x = (2 ± √16) ÷ 2 = (2 ± 4) ÷ 2.
Plus: 6 ÷ 2 = 3. Minus: −2 ÷ 2 = −1.
Factoring gives the same answers: (x − 3)(x + 1) = 0. The figure below starts here.
- a = 1, b = −2, c = −3
- b² − 4ac = (−2)² − 4(1)(−3) = 16
- x = (2 ± √16) ÷ 2 = −1 or 3
Example 2: an answer that is not a whole number
x² − 4x + 1 = 0: a = 1, b = −4, c = 1.
b² − 4ac = 16 − 4 = 12.
x = (4 ± √12) ÷ 2. Since √12 = 2√3, this is 2 ± √3.
That is about 3.73 and 0.27.
What the discriminant tells you
D > 0: two solutions. The parabola crosses the x-axis twice.
D = 0: one solution. x² − 4x + 4 = 0 has D = 16 − 16 = 0, so x = 4 ÷ 2 = 2. The parabola touches the x-axis once.
D < 0: no real solutions. x² + 2x + 5 = 0 has D = 4 − 20 = −16. You cannot take the square root of a negative number, and the parabola never reaches the x-axis.
Where the formula comes from
The vertex of a parabola is at x = −b ÷ 2a, the middle. The ± part, √D ÷ 2a, is how far the two solutions sit on each side of that middle. A bigger discriminant means the solutions are further apart. When D = 0 the distance is 0, so the two solutions meet on the vertex. This is the same “halfway between the roots” picture as in factored form.
Watch it explained in under a minute
The four mistakes to avoid
- Squaring a negative b the wrong way. (−2)² is 4, not −4.
- Forgetting that −b flips the sign of b. For b = −2, −b is 2.
- Dividing only the root by 2a. The whole top, −b ± √D, is divided by 2a.
- Forgetting that the equation must equal 0 before you read off a, b and c.
Try it yourself
Tap a question to see the answer.
1. x² − 6x + 5 = 0
D = 36 − 20 = 16. x = (6 ± 4) ÷ 2, so x = 5 or x = 1.
2. 2x² + 3x − 2 = 0
a = 2, b = 3, c = −2. D = 9 + 16 = 25. x = (−3 ± 5) ÷ 4, so x = ½ or x = −2.
3. How many real solutions does x² + 3x + 5 = 0 have?
D = 9 − 20 = −11, which is negative, so none. The parabola never reaches the x-axis.
Where this fits in
That finishes the quadratics part of the course: what a quadratic is, vertex form, factored form, factoring, and the formula. To see how the same ideas play out in the app, try quadratics, visually, or browse every post in the series index.