Monomial Equations
A monomial equation is a particular kind of quadratic equation of the form ax2 + bx + c = 0, where the coefficients b and c are both zero, and a ≠ 0. In this case, the equation simplifies to: $$ ax^2 = 0 $$
Equations of this type always have the same outcome: two identical real roots, both equal to zero.
$$ x_1 = x_2 = 0 $$
Example
Let's look at a simple example:
$$ 3x^2 = 0 $$
The equation holds true only when x = 0.
$$ 3(0)^2 = 0 $$
$$ 0 = 0 $$
So, the solution is straightforward:
$$ x = 0 $$
Step-by-step explanation
Now let's see why this is always the case. Start with the general form:
$$ ax^2 = 0 $$
To isolate x, divide both sides by the nonzero coefficient a:
$$ \frac{ax^2}{a} = \frac{0}{a} $$
$$ x^2 = 0 $$
Next, take the square root of both sides:
$$ \pm \sqrt{x^2} = \pm \sqrt{0} $$
The square root and the exponent cancel each other out on the left-hand side:
$$ \pm x = \pm \sqrt{0} $$
Since the square root of zero is zero (because 02 = 0), we have:
$$ \pm x = 0 $$
This gives two possible values for x:
$$ x = \begin{cases} +x = 0 \\ \\ -x = 0 \end{cases} $$
However, both lead to the same result. The equation has one real solution, repeated twice: x = 0.
This simple case is useful because it helps illustrate the properties of quadratic equations and how they behave when only one term remains.
