The Standard Form of a Quadratic Equation
When we use mathematics to model real world situations, we sometimes
use quadratic equations. Such equations are also called second degree
equations, or equations of degree 2.
A quadratic equation in one variable is an equation that can be
written in the form
ax2 + bx +c = 0
where a, b, and c are real numbers and a ≠ 0.
This is called the standard form of a quadratic equation.
A quadratic equation must have a term of
degree 2, such as x2.
It cannot have a term of higher degree.
In the definition, notice that the terms on the left side of the equation are
arranged in descending order by degree. The right side of the equation is
We can also write a quadratic equation
with 0 on the left side, like this:
0 = ax2 + bx + c
Here are some examples of quadratic equations. To determine the values
of a, b, and c, we first write the equation in standard form, ax2 + bx + c
x2 - 3x = 28
5x2 = -45
-3x2 = -12x
1x2 - 3x - 28 = 0
5x2 + 0x + 45 = 0
-3x2 + 12x + 0 = 0
a = 1, b = -3, c= -28
a = 5, b = 0, c = 45
a = -3, b = 12, c = 0
The variable in a quadratic equation can
be any letter, not just x.
3y2 + 5y - 9 = 0 is a quadratic equation.
For each of the following, if the equation is quadratic, write it in standard
form and identify the values of a, b, and c.
a. 3x(x + 4) = 18
b. -6x2 = 8 - 3x(2x + 1)
c. 0 = 7x2
Subtract 18 from both sides.
This is a quadratic equation in standard form.
Here a = 3, b = 12, and c = -18.
| 3x(x + 4)
3x2 + 12x
3x2 + 12x - 18
Add 6x2 to both sides and rearrange terms.
Because the coefficient of the
x2-term is 0, the equation is not
a quadratic equation.
|= 8 - 3x(2x + 1)
= 8 - 6x2 - 3x
= 0x2 - 3x + 8
= -3x + 8
Fill in the missing x-term and the
missing constant term.
This is a quadratic equation,
ax2 + bx + c = 0, where a = 7, b = 0, and c = 0.
= 7x2 + 0x + 0