Quadratic Expressions

A quadratic expression in one variable has highest exponent 2 and can be written as $ax^{2} + bx + c$ with $a \ne 0$ . It is an expression until it is set equal to another quantity.

Different equivalent forms reveal different information. Standard form shows coefficients; factored form shows potential zeros; completed-square form shows the vertex and minimum or maximum when the expression defines a function.

On this page (8 sections)

Standard form and its parts

ax^2+bx+c,\qquad a\ne0

The coefficient $a$ is the leading coefficient, $b$ is the linear coefficient and $c$ is the constant term. Here $a$ , $b$ and $c$ are real constants. A missing linear or constant term simply has coefficient zero.

For example, $4x^2+7$ is quadratic with $b=0$ . The expression $0x^2+4x+7$ is linear, not quadratic. Identify degree after simplifying, since apparently quadratic terms can cancel.

Evaluating a quadratic expression

For $x^2-4x+3$ at $x=5$ , substitute into every occurrence of the variable.

5^2-4(5)+3=25-20+3=8

If the input were $-1$ , parentheses would give $(-1)^2-4(-1)+3=1+4+3=8$ . Different inputs can yield the same value.

Equivalent forms reveal different structure

Three equivalent forms of the same quadratic
Form	Expression	Information visible
Standard	$x^2+x-6$	Leading, linear and constant coefficients
Factored	$(x-2)(x+3)$	Zeros at $2$ and $-3$ when the expression is set equal to zero
Completed square	$(x+\frac12)^2-\frac{25}{4}$	Minimum value $-25/4$ attained at $x=-1/2$

(x-2)(x+3)=x^2+x-6

\left(x+\frac12\right)^2-\frac{25}{4}=x^2+x+\frac14-\frac{25}{4}=x^2+x-6

Both expansions return the standard form, so these are three representations of the same expression. The most useful representation depends on whether you need coefficients, zeros or a minimum.

Expression, equation and function are different roles

$x^2-4x+3$ is an expression. The statement $x^2-4x+3=0$ is an equation, whose real solutions are $1$ and $3$ . The rule $f(x)=x^2-4x+3$ , together with a domain, defines a function.

The algebra is related, but the tasks differ: evaluate or rewrite the expression, solve the equation, or describe the outputs and graph of the function.

Connection to quadratic functions

For the function $f(x)=x^2-4x+3$ , completing the square gives $f(x)=(x-2)^2-1$ . Its graph is a parabola with vertex $(2,-1)$ and symmetry line $x=2$ .

Read the zeros and the minimum from the same quadratic

$y=x^2-4x+3=(x-2)^2-1$

The zeros are

1

and

3

; the minimum point is

(2,-1)

Diagram explained

The parabola is symmetric about x = 2. Outputs decrease to −1 and then increase. The two roots lie one unit on either side of the symmetry line.

Exact reference values for this graph
Input	Output
$0$	$3$
$1$	$0$
$2$	$-1$
$3$	$0$
$4$	$3$

Since $(x-2)^2\ge0$ , the least output is $-1$ , attained at $x=2$ . The roots $1$ and $3$ sit equally far from the symmetry line. These features connect the expression to its graph without requiring calculus.

Practice questions

Is $4x^{2} – x + 7$ quadratic?
Evaluate $x^{2} – 4x + 3$ at $x = 5$ .
Expand $(x – 2)(x + 3)$ .

Check your answers

Yes. In $4x^2-x+7$ , the highest power is 2 with non-zero coefficient 4.
$5^2-4(5)+3=25-20+3=8$ .
$(x-2)(x+3)=x^2+3x-2x-6=x^2+x-6$ .

Sources and further reading

OpenStax: Quadratic forms and graphs. Definitions, conventions and related examples.
OpenStax Elementary Algebra 2e. Language of algebra, equations, inequalities and factoring.
OpenStax Intermediate Algebra 2e. Quadratic equations, factoring and algebraic techniques.
Wolfram Language: Algebraic Transformations. Symbolic simplification, expansion and factoring reference.