The Three Forms and What Each One Reveals

In this lesson you will: Move between standard, factored and vertex form for the same quadratic function; Complete the square on a quadratic with leading coefficient 1 to expose the vertex; Choose the form that answers the question you were actually asked.

One function, three outfits

A quadratic can be written three ways, and each one puts a different feature in plain sight. Standard form ax^2+bx+c shows the y-intercept. Factored form a(x-r)(x-s) shows the x-intercepts. Vertex form a(x-h)^2+k shows the vertex.

None of them is the \"real\" one. They are the same function, and expanding or factoring moves you between them without changing a single output.

Completing the square, with a leading coefficient of 1

You met this in Chapter 15 as a way to solve. Here it does a different job: it rewrites a function so the vertex can be read off.

Halve the coefficient of x, square it, then add and subtract that number. For x^2+10x+18, half of 10 is 5 and 5^2=25, so x^2+10x+18=(x^2+10x+25)-25+18=(x+5)^2-7. The vertex is (-5,-7).

Ohio's Algebra 1 work with this standard concentrates on leading coefficient 1, which is exactly the case where the halving is clean.

Pick the form that matches the question

Most of the difficulty students have with this standard is not algebra. It is starting the wrong conversion — completing the square when the question asked for the zeros, or factoring when it asked for the minimum value.

Common mistake — Losing the number you added when completing the square

Completing the square on x^2+10x+18, a student writes (x+5)^2+18.

The reasoning is half right. Half of 10 is 5, so (x+5)^2 is correct — but (x+5)^2 is x^2+10x+25, which is 25 more than the original quadratic's first two terms. The 25 has to come back out.

The correct line is (x+5)^2-25+18=(x+5)^2-7. The student's version is a different function, sitting 25 units higher.

How to catch it: Test one input in both versions — x=0 is the easiest. The original gives f(0)=18; the correct form gives 5^2-7=18; the faulty one gives 5^2+18=43. Any disagreement means the constant was not adjusted.