The quadratic formula
The roots are x = (−b ± √(b² − 4ac)) / 2a. The part under the root, b² − 4ac, is the discriminant.
For x² − 5x + 6 the discriminant is 25 − 24 = 1, so the roots are (5 ± 1) / 2, which are 3 and 2.
What the discriminant decides
Positive: two different real roots, so the curve crosses the x-axis twice.
Zero: one repeated root, so the curve just touches the axis. Negative: no real roots, so the curve never reaches it.
Worked example
x² − 5x + 6 = 0 has roots x = 3 and x = 2, a discriminant of 1 and a vertex at (2.5, −0.25).
It factorises as (x − 2)(x − 3), which is the same answer written another way: each bracket gives one root.
Roots that are not real
When the discriminant is negative the curve sits entirely above or below the x-axis, so no real number solves the equation.
The calculator writes the answer with an i instead of refusing the input, so you can see the complex pair.
Inputs for the solving quadratics calculator
Coefficient of x²
The number multiplying x². It cannot be zero, or the equation is linear.
Coefficient of x
The number multiplying x, including its sign.
Constant
The number on its own. A zero here means one root is zero.
CALCULATOR QUESTIONS
Solving quadratics calculator questions
How do I solve x² − 5x + 6 = 0?
The discriminant is 1, so the roots are (5 ± 1) / 2, which are 3 and 2.
What is the discriminant?
b² − 4ac. Its sign tells you whether there are two real roots, one repeated root, or none.
Why does the vertex matter?
It is the turning point of the curve. Its x value is −b / 2a, halfway between the two roots.
What if the roots are not whole numbers?
They are still shown, as decimals and in exact surd form such as (3 ± √5) / 2.