Definition:

A quadratic equation in the variable x is an equation that can be written in the form:

where a, b and c represent real number coefficients.

This form is sometimes called the standard form. The term quadratic is used for any equation where the highest power of the variable x is 2. The coefficient a cannot be zero, since otherwise it would be a linear equation.

Formula for solving quadratic equations (known as the quadratic formula):

To solve quadratic equations, substitute the coefficients a, b and c into the quadratic formula.

The expression b² – 4ac shown under the square root sign is called the discriminant, because it can "discriminate" between the all possible types of answer:

type 1: If b² - 4ac ≥ 0 ⇒ equation has two real roots;

type 2: If b² - 4ac = 0 ⇒ equation has two real roots but they are both the same.

type 3: If b² - 4ac ≤ 0 ⇒ equation has two complex roots;

Example 1:

Solve the following quadratic equation using the quadratic formula.

Solution:

In this case a=2 b=7 c=–15

The value of the discriminant is b² – 4ac = 72 – 4(2)(–15) = 169 (Type 1)

Exercise 1: Solve quadratic equations

Level 1

Level 2

Example 2:

Solve the following equation using the quadratic formula.

Solution:

In this case a = 4 b = – 20 c = 25

The value of the discriminant is b² – 4ac = 202 – 4(4)(25) = 0

That is, in this case since the value of the discriminant is zero, the two roots of the equation have the same of 2.5.

Exercise 2: Solve quadratic equations

Level 1

Level 2

Example 3:

Solve the following quadratic equation using the quadratic formula.

Solution:

In this case a = 5 b = 2 c = 3

The value of the discriminant is b² – 4ac = 42 – 4(5)(3) = – 44

Since the value of the discriminant is negative, this equation has no roots that are real numbers.