Advanced Math / Equivalent Expressions

Equivalent Quadratic Forms

Quadratic expressions can be written in standard form, factored form, or vertex form. Moving between these forms reveals different information, such as zeros, coefficients, or the vertex.

Least you need to know

To expand factored form, multiply each term in one factor by each term in the other factor.
To factor a monic quadratic $x^2+bx+c$, look for two numbers with sum $b$ and product $c$.
The square $(x+h)^2$ expands to $x^2+2hx+h^2$.
Vertex form $a(x-h)^2+k$ can be expanded by first expanding the square, then distributing $a$.

Key notation

$(x-3)(x+7)$ — Factored form of a quadratic expression
$(x-4)^2-4$ — Vertex form of a quadratic expression

Worked example

To rewrite $x^2-8x+12$ in vertex form, complete the square.
$x^2-8x+12=(x^2-8x+16)-4$.
Therefore, $x^2-8x+12=(x-4)^2-4$.

Common mistakes

When expanding $(x-a)^2$, the middle term is $-2ax$, not $-ax$.
The constant term in factored form is the product of the constants.
A negative leading coefficient changes whether the vertex is a maximum or minimum.

How to recognize it

The problem asks which expression is equivalent.
The expression is given in factored, standard, or vertex form.
The problem asks for a missing coefficient or parameter in an equivalent quadratic.

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