The quadratic structure explorer takes the coefficients of y = ax² + bx + c and shows everything that belongs together: the discriminant, the roots, the turning point and three linked forms of the same expression. A graph and a table of values sit underneath.
Use it after you have tried a question by hand, to check your work and see the connections.
How do you use it?
- Enter a, the coefficient of x². The tool only treats the expression as a quadratic when a is not 0.
- Enter b, the coefficient of x, and c, the constant.
- Set the graph range: Graph from x = and to x =. The left value must be smaller than the right value.
- Press Explore.
- Press Reset to go back to the sample, y = x² − 5x + 6.
Negative numbers are fine. Enter them with a minus sign, for example −5.
How do you read the result?
The first line gives the discriminant b² − 4ac and what its sign means. Then you see the roots, marked exact or rounded, and the turning point with a note saying whether it is a minimum or a maximum. A positive a gives a minimum, and a negative a gives a maximum.
Under Linked forms you get the standard form, the factorised form and the completed-square form. Read them together: the factors show the roots, and the completed-square form shows the turning point.
The graph marks the roots and the turning point when they fall inside your range. If a point is missing, widen the range. The table of values gives the same information in text.
Example walk-through
The sample is y = x² − 5x + 6. The discriminant is 25 − 24 = 1, which is positive, so there are two real roots.
- Roots: 2 and 3.
- Turning point: (5/2, −1/4), a minimum.
- Standard form: x² − 5x + 6.
- Factorised form: (x − 2)(x − 3).
- Completed-square form: (x − 5/2)² − 1/4.
Notice that the roots 2 and 3 have a midpoint of 5/2, which is the x-coordinate of the turning point. That is no accident.
Now try a = 1, b = 2, c = 5. The discriminant is 4 − 20 = −16, so there are no real roots and the tool says factorised form is not available. The completed-square form is (x + 1)² + 4, so the lowest point is (−1, 4), which sits above the x-axis.
For x² − 2x − 1 the discriminant is 8, and the roots are 1 ± √2. The tool shows the exact form and the rounded values, about −0.4142 and 2.4142.
What are the assumptions and limits?
- Only real roots are found. A negative discriminant means none.
- When a = 0 the tool reports a straight line instead of a quadratic.
- Exact answers are fractions or surds. Rounded answers are marked ≈, and the factorised form then uses rounded roots.
- The graph is drawn only for the range you choose, so a root or turning point outside it will not appear.
- The tool uses your numbers as given. It cannot tell whether you copied a question correctly.
Which lessons explain the ideas behind it?
- Factorise a quadratic with integer factors is the place to start for the factorised form.
- Complete the square without losing a coefficient builds the completed-square form step by step.
- Use the discriminant to classify intersections explains the three cases you see in the first line.
- Recover a quadratic from roots works backwards from the roots to the expression.
- Connect a turning point with a minimum value links the vertex to the highest or lowest value.
The wider topic is quadratic structure and discriminants. A teacher can also help you choose which form to use under exam conditions, for example in online one-to-one Additional Mathematics tuition. Other tools are in the learning tools directory.