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Quadratic Equations

What is a 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,

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

is called quadratic equation in x.

Quadratic Equation Example:

Quadratic Equation

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:

2x2 + 5x + 3 = 0 In this one a=2, b=5 and c=3
x2 − 3x = 0 This one is a little more tricky:

  • Where is a? Well a=1, as we don’t usually write 1x2
  • b = −3
  • And where is c? Well c=0, so is not shown.

What is the quadratic formula?

The quadratic formula is a way to solve any quadratic equation. If ax2 + bx + c = 0, then you can solve for x using the quadratic formula, which is:

quadratic formula

How do I solve a quadratic equation?

Using the quadratic formula,
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:
quadratic formula
Put in a, b and c: x = −6 ± √(6− 4×5×1)/2×5
Solve:x = −6 ± √(36 − 20)/10
 x = −6 ± √(16)/10
 x = −6 ± 4/10
 x = −0.2 or −1