Standard and Factored Forms of Quadratic Expressions

7.9.3 Standard and Factored Forms of Quadratic Expressions

7.9.3 • Standard and Factored Forms of Quadratic Expressions

Activity

The quadratic expression $x^{2} + 4 x + 3$ is written in standard form.

Here are some other quadratic expressions. The expressions on the left are written in standard form and the expressions on the right are not.

Written In Standard Form	Not Written In Standard Form
$x^{2} - 1$	$(2 x + 3) x$
$x^{2} + 9 x$	$(x + 1) (x - 1)$
$\frac{1}{2} x^{2}$	$3 (x - 2)^{2} + 1$
$4 x^{2} - 2 x + 5$	$- 4 (x^{2} + x) + 7$
$3 x^{2} - x + 6$	$(x + 8) (- x + 5)$

What are some characteristics of expressions in standard form?

Compare your answer:

The standard form always starts with an $x^{2}$ term and is in decreasing order of degree.
It does not need to have an $x$ term or a numerical term.
Standard form is $a x^{2} + b x + c$ where $a$ is not zero.
The standard form shows a sum or difference of terms.

$(x + 1) (x - 1)$ and $(2 x + 3) x$ in the right column are quadratic expressions written in factored form. Why do you think that form is called factored form?

Compare your answer:

It is a product of two factors. Each factor could be a variable, a number, or an expression with a variable (with no exponent) and a number.

Are you ready for more?

Extending Your Thinking

Which quadratic expression can be described as being both standard form and factored form? Explain how you know.

Compare your answer:

$x^{2}$ could be viewed as both standard and factored. It has an $x^{2}$ term even though there is not a constant or linear term. It can also be written as the product of two linear expressions because $x^{2}$ is equal to $x \cdot x$ .

Self Check

Which of the following expressions is a quadratic expression written in standard form?

$7(x-6)^2+1$
$x(x-5)$
$x^2-9x+20$
$x-4$

Additional Resources

Standard Form of Quadratics

Examples of Standard Form Quadratics

$x^{2}$

$x^{2} - 9$

$x^{2} - 3 x$

$x^{2} + 4 x - 45$

These examples have an $x^{2}$ and lead with it. Some have an $x$ term or a constant term but it is not needed to be in standard form. Standard form of a quadratic expression is of the form $a x^{2} + b x + c$ , where $a \neq 0$ .

Non-Examples of Standard Form Quadratics

$7 - x$

$9 (x - 2)$

$x (x + 10)$

These examples do not have an $x^{2}$ term.

Try it

Standard Form of Quadratics

Determine if $4 x (x - 2)$ is a quadratic in standard form.

Here is how to determine if the expression is in standard form:

This expression is in factored form. If the $4 x$ is distributed, the expression becomes $4 x^{2} - 8 x$ , which is in standard form.