Rewriting Expressions and Equations in Equivalent Forms
The same expression can be written in different equivalent forms, each built to show off one fact fast. Rewriting is just repackaging the information, not changing it.
A quadratic like x2 - 6x + 5 can be standard form, factored form, or vertex form. Standard form is easy to generate but hides its best features. Factored form, (x-1)(x-5), hands you the roots immediately: x = 1 and x = 5, since each factor can equal zero. Vertex form, (x-3)2 - 4, hands you the lowest point of the graph directly: the vertex is (3, -4), because a squared term is always zero or positive, so the expression is smallest when (x-3)2 = 0.
The mechanism behind vertex form is completing the square. Take half of the x-coefficient, square it, and add and subtract that amount: x2 - 6x + 5 = (x2 - 6x + 9) - 9 + 5 = (x-3)2 - 4. That same half-and-square move is what the quadratic formula automates, and it is what factoring reverses when the roots are nice integers. All three paths - factoring, completing the square, and the formula - describe the same parabola from different angles.
On a test, an answer choice rarely looks like the form you naturally reach. If you solve and get 2(x+3)2 but none of the four choices resemble that, don't assume you made an error. Expand your result and compare it to each choice in standard form, or plug in a convenient number like x = 0 to both your expression and each choice. Equivalent expressions are allowed to look completely different while producing identical outputs for every input, which is exactly what makes this family of questions - linear, quadratic, rational, radical, polynomial rewriting - connect: they all test whether you can move between representations without losing information.
Key terms
- Standard form
- A quadratic written as ax2 + bx + c, convenient for reading the constant term and leading coefficient.
- Factored form
- A quadratic written as a product like (x-p)(x-q), which directly reveals the roots p and q.
- Vertex form
- A quadratic written as a(x-h)2 + k, which directly reveals the vertex (h, k), the graph's minimum or maximum.
- Completing the square
- Adding and subtracting half the x-coefficient squared to rewrite a quadratic as a perfect square plus a constant.
- Equivalent expressions
- Expressions that produce the same output for every input even though they are written in different forms.
Watch it explained
- How it worksHow does factored form hand over the roots while vertex form hands over the lowest point?
- Worked exampleWalk me through rewriting x2 - 6x + 5 in vertex form and reading off its minimum.
- Common mistakeWhy isn't your answer wrong just because it looks nothing like any of the four choices at first?
- Ties togetherHow does picking the right equivalent form tie factoring, completing the square and the formula together?
Check yourself
End of topic.