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.
Definition â€”
Quadratic Equation
A quadratic equation in one variable is an equation that can be
written in the form
ax^{2} + bx +c = 0
where a, b, and c are real numbers and a ≠ 0.
This is called the standard form of a quadratic equation.
Note:
A quadratic equation must have a term of
degree 2, such as x^{2}.
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
zero.
We can also write a quadratic equation
with 0 on the left side, like this:
0 = ax^{2} + bx + cHere are some examples of quadratic equations. To determine the values
of a, b, and c, we first write the equation in standard form, ax^{2} + bx + c
= 0.
Nonstandard form
x^{2}  3x = 28
5x^{2} = 45
3x^{2} = 12x 
Standard form
1x^{2}  3x  28 = 0
5x^{2} + 0x + 45 = 0
3x^{2} + 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.
For example:
3y^{2} + 5y  9 = 0 is a quadratic equation.
Example 1
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. 6x^{2} = 8  3x(2x + 1)
c. 0 = 7x^{2}
Solution
a. 
Distribute 3x.
Subtract 18 from both sides.
This is a quadratic equation in standard form.
Here a = 3, b = 12, and c = 18. 
3x(x + 4)
3x^{2} + 12x
3x^{2} + 12x  18 
= 18
= 18
= 0 
b. 
Distribute 3x.
Add 6x^{2} to both sides and rearrange terms.
Because the coefficient of the
x^{2}term is 0, the equation is not
a quadratic equation. 
 6x^{2}
 6x^{2}
0
0 
= 8  3x(2x + 1)
= 8  6x^{2}  3x
= 0x^{2}  3x + 8
= 3x + 8 
c. 
Fill in the missing xterm and the
missing constant term.
This is a quadratic equation,
ax^{2} + bx + c = 0, where a = 7, b = 0, and c = 0. 
0
0 
= 7x^{2} = 7x^{2} + 0x + 0 
