The three forms of a quadratic
A quadratic can be written three ways, and each way makes one fact obvious while hiding the others. The skill is reading the question, deciding which fact it wants, and choosing the form that gives that fact without algebra.
Standard form: the y intercept and the direction
Written as y = ax² + bx + c, the constant c is the y intercept, because substituting zero for x leaves c. The sign of a tells you which way the parabola opens: positive opens upward, so the vertex is a minimum, and negative opens downward, so the vertex is a maximum. Two facts, no work.
Factored form: the roots
Written as y = a(x − r)(x − s), the numbers r and s are the solutions, which are also the x intercepts. Set each bracket to zero and solve. Factored form is the fastest route whenever the question asks for zeros, solutions, roots or intercepts, and the vertex sits exactly halfway between the two roots, so it also gives you the axis of symmetry.
Vertex form: the maximum or minimum
Written as y = a(x − h)² + k, the vertex is the point (h, k). Watch the sign: (x − 3)² puts the vertex at 3, while (x + 3)² puts it at -3. When a question asks for the maximum value, the minimum value, the vertex or the axis of symmetry, the vertex form is already the answer, and converting to it should be the last resort rather than the first.
Choosing the form before you calculate
Read the last line of the question, name the fact it asks for, then match it to a form. This takes ten seconds and saves the two minutes usually spent expanding something that never needed expanding.
- Maximum or minimum, vertex, axis of symmetry: vertex form
- Solutions, roots, zeros, x intercepts: factored form
- y intercept, or the value at x equals zero: standard form
- How many solutions: the discriminant, which needs none of the three
Converting when the form is wrong
From standard form, the vertex x coordinate is -b divided by 2a, and substituting that value back gives the y coordinate. From factored form, the vertex x coordinate is the midpoint of the two roots. Completing the square is the general method for writing standard form as vertex form, and it is worth doing once by hand so the shortcut stops feeling like a trick.
Sum and product of roots
For ax² + bx + c = 0, the two roots add to -b/a and multiply to c/a. Questions asking for the sum of the solutions, or for a constant given a relationship between the roots, are usually one line with these two facts rather than two separate solves.
How do I find the vertex without vertex form?
Use x equals -b divided by 2a for the x coordinate, then substitute that x into the equation for the y coordinate. In factored form, the x coordinate is the midpoint of the two roots.
How do I know which form a question wants?
Read what it asks for. Vertex or maximum points to vertex form, roots or x intercepts point to factored form, and the y intercept points to standard form.
What does the discriminant tell me?
The sign of b² - 4ac decides how many real solutions exist, without solving anything. There is a separate guide on the discriminant.
Why is the vertex halfway between the roots?
A parabola is symmetric about its vertical axis, and the two roots sit at equal distances from that axis, so their midpoint is the x coordinate of the vertex.
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