Quadratic Equation Without Constant

Another common type of a quadratic equation is a quadratic equation without any constant. That is, each equation

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

is a quadratic equation without constant. The approach how to find roots of such equation is easy. Let’s take an example

8x2 − 4x = 0.

We can factor out the variable x:

8x2 − 4x = 0
x · (8x − 4) = 0

We can immediately see one of the roots: it’s the x1 = 0. Because if we substitute x for zero, the equation definitely holds since

0 · (8 · 0 − 4) = 0.

Ok, now the expression inside parenthesis. The whole left part of the equation equals zero when the inside of the parenthesis equals zero. Thus we’re actually solving linear equation 8x − 4 = 0. We can find the root of this equation easily

\begin{eqnarray}8x - 4 &=& 0\\\\8x&=&4\\\\x&=&\frac{4}{8}\\\\x&=&\frac{1}{2}\end{eqnarray}

The second root is x2 = ½. Let’s take a look at the graph:

Graph of a function 8x^2-4x.svg

The graph of quadratic function f(x) = 8x2 − 4x intersects the horizontal axis in two points: (0, 0) and (½, 0). It’s exactly what we’ve solved.