Quadratic Forms: Standard, Vertex and Factored Form

A quadratic function can be written in three forms: standard form ax² + bx + c, vertex form a(x − h)² + k and factored form a(x − x₁)(x − x₂), and each form shows different information. In brief: the function f(x) = x² − 6x + 5 equals (x − 3)² − 4 (vertex form, vertex (3, −4)) and (x − 1)(x − 5) (factored form, zeros 1 and 5).

Choosing the right form makes problems easy: standard form gives the y-intercept, vertex form gives the maximum or minimum and factored form gives the zeros. This guide shows what each form tells you and how to convert between them with worked examples, summarised in a comparison table.

What does standard form tell you?

In f(x) = ax² + bx + c, the coefficient a gives the direction and width of the parabola, and c is the y-intercept, since f(0) = c. The coefficient b does not have such a simple meaning. It affects the position of the axis of symmetry, x = −b/(2a). For f(x) = 2x² − 8x + 5, the axis is x = 8/4 = 2 and the y-intercept is (0, 5).

What does vertex form tell you?

In f(x) = a(x − h)² + k, the vertex is (h, k) and the axis is x = h. If a > 0 the minimum value is k, and if a < 0 the maximum value is k. The graph is a shifted and stretched copy of y = x².

Example: f(x) = −3(x + 2)² + 12 has vertex (−2, 12), opens downward and has maximum value 12. The sign in the bracket is opposite to the shift: x + 2 means h = −2.

What does factored form tell you?

In f(x) = a(x − x₁)(x − x₂), the numbers x₁ and x₂ are the zeros, and the axis of symmetry is halfway between them: x = (x₁ + x₂)/2. Not every quadratic function has a factored form with real numbers; it exists only when the graph meets the x-axis.

Example: f(x) = 2(x + 1)(x − 5). The zeros are −1 and 5, the axis is x = 2, the vertex is (2, f(2)) = (2, 2 · 3 · (−3)) = (2, −18) and the y-intercept is f(0) = 2 · 1 · (−5) = −10.

How do you convert between the forms?

Factored to standard: expand

(x − 1)(x − 5) = x² − 6x + 5. Multiply the brackets and collect terms.

Vertex to standard: expand the square

2(x − 3)² − 4 = 2(x² − 6x + 9) − 4 = 2x² − 12x + 14.

Standard to vertex: complete the square

1. Factor a out of the x-terms: a(x² + (b/a)x) + c.
2. Add and subtract (b/(2a))² inside the bracket.
3. Write the perfect square and collect constants.

Example: f(x) = 2x² − 12x + 14 = 2(x² − 6x) + 14 = 2(x² − 6x + 9 − 9) + 14 = 2(x − 3)² − 18 + 14 = 2(x − 3)² − 4.

Standard to factored: solve f(x) = 0

Find the zeros by factoring or the quadratic formula. For 2x² − 12x + 14 = 0, divide by 2: x² − 6x + 7 = 0, so x = 3 ± √2. The factored form is 2(x − 3 − √2)(x − 3 + √2). For 2(x − 3)² − 4 = 0: (x − 3)² = 2, which gives the same zeros ✓.

Vertex from the standard form directly

h = −b/(2a) and k = f(h). For f(x) = x² − 6x + 5: h = 3, k = 9 − 18 + 5 = −4. So the vertex is (3, −4).

Comparison of the three forms

Check the equality at x = 0: 5 = 9 − 4 = (−1)(−5) = 5 ✓.

Where do these forms come from?

Completing the square was known to the Babylonians, in the form of procedures for solving quadratic problems, and was described by al-Khwarizmi in the ninth century. The factored form relates to the idea of roots as linear factors. The vertex form, as a way to read a maximum or minimum without calculus, became standard in textbooks of the twentieth century.

When is each form useful in practice?

For the height of a ball, h(t) = −4.9t² + 20t + 1.5, the standard form gives the initial height 1.5 m. Vertex form shows the maximum height and the time at which it occurs. The factored form shows when the ball is at ground level. Which form to pick depends on the question.

Which mistakes are common?

Wrong sign of h

(x + 4)² has h = −4, not 4.

Forgetting to factor out a before completing the square

For 2x² − 12x, first write 2(x² − 6x). Otherwise the constant is wrong.

Losing the leading coefficient in factored form

2x² − 8 = 2(x − 2)(x + 2), not (x − 2)(x + 2).

Assuming every quadratic factors over the reals

x² + 1 has no real zeros, so there is no real factored form.