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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 — Quadratic Equation

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 zero.

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 = 0.

Non-standard form

x2 - 3x = 28

5x2 = -45

-3x2 = -12x

Standard form

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.

For example: 3y2 + 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. -6x2 = 8 - 3x(2x + 1)

c. 0 = 7x2



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)

3x2 + 12x

3x2 + 12x - 18

= 18

= 18

= 0


Distribute -3x.

Add 6x2 to both sides and rearrange terms.

Because the coefficient of the x2-term is 0, the equation is not a quadratic equation.

- 6x2

- 6x2



= 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

= 7x2 + 0x + 0

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