Quadratic Equations

Definition of Quadratic Equation

A quadratic equation is an equation that can be written in a form like this: ax² + bx +c = 0. In this equation, a, b, and c are constant numbers. The numbers a and b are called coefficients because they are multiplied with x. X is a variable. The equation can be written in different ways, such as ax² = -bx – c, but simply rearranging the equation doesn’t change the fact that it is a quadratic equation.

Standard Form

An  equation of the form ax² + bx +c = 0,

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• where a,b,c  are real numbers
• and a ≠ 0

is called a quadratic equation in x.

Quadratic Equation Example:

Root of a Quadratic Equation: A real number k is called a root of the Quadratic Equation ax² + bx +c = 0, a ≠ 0 If ak² + bk +c = 0.

Note 1: if k is a root of ax² + bx +c = 0, then we say that

• X=k satisfies the equation ax² + bx +c = 0 or
• X=k is a solution of the equation ax² + bx +c = 0.

Note 2: The roots of a Quadratic Equation ax² + bx +c = 0 are called the zeros of the polynomial ax² + bx +c .

Here are some examples:

Quadratic formula

How do I solve a quadratic equation?

Using the quadratic formula,

you can solve for x if you know the values of a, b and c.

When you solve a quadratic equation with the quadratic formula, you will always find two solutions. That is because there are always two values of x that satisfy the conditions of the quadratic formula. So you will never find exactly one solution. However, not all solutions are real numbers. When you calculate your solution to your quadratic equation, you might find that you have:

• two real number solutions,
• or, two solutions, both of which are complex numbers.

Example: find the roots of the following equation 5x² + 6x + 1 = 0

Coefficients are:a = 5, b = 6, c = 1

Quadratic Formula:

Put in a, b and c: x = −6 ± √(6² − 4×5×1)/2×5

Solve:x = −6 ± √(36 − 20)/10

 x = −6 ± √(16)/10

 x = −6 ± 4/10

 x = −0.2 or −1