Quadratic Equation

A quadratic equation is any equation in the form

ax2 + bx + c = 0,      where a, b, c ∈ ℝ and a≠0.

The important part is the ax2. Every quadratic equation has to have this term and a ≠ 0. Because if a = 0 then we can simplify the equation to bx + c = 0 and that’s actually a linear equation. The a, b and c are real numbers and they are usually called quadratic coefficient, linear coefficient and constant, respectively.

These are examples of quadratic equations:

  • 3x2 + 4 − 1 = 0.
  • − x2 + 42 = 0.
  • 10x2 + x = 0.
  • x2 = 0.

These are examples of equations that are not quadratic:

  • 3x + 6 = 0 because there is no x2.
  • x3 + x2 = 0 because these is x3. Only x2 powers are allowed.
  • 2x2 + 3x − sin(x) = 0 because of presence of sinus.

Identifying The Coefficients

When solving any quadratic equation, it’s kind of crucial to correctly identify each coefficient. Let’s try it. Let’s take the four examples of quadratic equations and we’ll try to identify the coefficients:

  • 3x2 + 4 − 1 = 0 This one is easy. The quadratic coefficient a is clearly equals to a = 3. The linear coefficient is equal to b = 4 and finally the constant equals to − 1. Just be aware that the minus sign is part of the coefficient. If it helps, imagine the equals is written this way: 3x2 + 4 + ( − 1) = 0.

  • − x2 + 42 = 0 This one seems more complicated. What is the quadratic coefficient? There is no number, just the minus sign. The − x is the same as − 1 · x thus the quadratic coefficient equals a = − 1. What about the linear coefficient? The is no linear coefficient so it’s the same as 0 · x which means the linear coefficient equals zero b = 0. The constant is easy, it’s c = 42.

  • 10x2 + x = 0 Ok, the quadratic coefficient is just ten, a = 10, no big deal. The linear coefficient is one since x = 1 · x, i. e. b = 1. And the constant is entirely missing, thus c = 0.

  • x2 This one looks weird but it really is an quadratic equation. The quadratic coefficient equals a = 1 and all other coefficients are missing, i. e. they equal zero. b = 0 and c = 0.

Quadratic Function

We can write a different definition of the quadratic equation. It’s every equation in the form

f(x) = 0,

where f(x) is a quadratic function. When solving such equation we’re actually finding the coordinates of all intersection points – when the graph of the function intersects the horizontal axis. From the shape of the graph – which is always a hypoerbola – we can see, that there can be up to two such points.

Let’s take for example the function f(x) = x2 − 4.

Graph of function y=x^2-4

We can see, from the picture above, that the graph intersects the horizontal axis in two points: ( − 2, 0) and (2, 0). It means that the equation f(x) = 0, or x2 − 4 = 0 in other words, would have two results: x1 = − 2 and x2 = 2. It’s the x-coordinate of the intersection points.

Now let’s look how to solve different type of quadratic equations.