Animated Solution for Mathematics - Quadratic Equations: Let $a>0,b>0$ and $c>0$. Then the roots of the equation $a{x}^2+bx+c=0$

Unlocking the Quadratic Equation

Welcome, future engineers and scientists. Today, we are going to dissect a problem that seems simple on the surface but hides a profound truth about the nature of quadratic equations.

We are looking at the equation $a{x}^2+bx+c=0$, where $a,b,c>0$. Many students rush to the quadratic formula, but I want you to pause and visualize. Mathematics is not just about calculation; it is about seeing the structure of the universe.

The Geometry of the Smile

Let us start by visualizing the function $f(x)=a{x}^2+bx+c$. Because the leading coefficient $a$ is strictly positive, we know immediately that this parabola opens upwards. It is a smile.

Now, consider the y-intercept. If we set $x=0$, we get $f(0)=c$. Since $c>0$, the parabola cuts the y-axis at a positive height. This is our starting point: a smile that begins above the origin.

The Vertex's Secret

Now, where is the vertex? The x-coordinate of the vertex of a parabola $a{x}^2+bx+c$ is given by the formula:

Look at this expression carefully. We are given that $a>0$ and $b>0$. Therefore, the ratio $\frac{b}{2a}$ is positive.

When we put a negative sign in front of it, the result is strictly negative. This tells us something powerful: the vertex of our parabola is always located to the left of the y-axis. Our 'smile' is shifted into the negative x-region.

The Real Root Scenario

What happens if the discriminant $D=b^2-4ac$ is greater than or equal to zero? This means the parabola intersects the x-axis. The roots are given by:

We need to determine the sign of these roots. We know that $D=b^2-4ac$. Because $a$ and $c$ are positive, $4ac$ is positive, which means $D<b^2$. Taking the square root, we find that $\sqrt{D}<b$.

Now, look at the roots:

Since $\sqrt{D}<b$, the numerator $-b+\sqrt{D}$ is still negative. And obviously, $-b-\sqrt{D}$ is even more negative. Thus, if the roots are real, they are both strictly negative. The parabola crosses the x-axis on the left side of the y-axis.

The Complex Root Scenario

But what if $D<0$? This is where many students panic. If $D<0$, the parabola floats above the x-axis and never touches it.

The roots are complex conjugates:

Here, the real part of the root is exactly $-\frac{b}{2a}$. We already established that $-\frac{b}{2a}$ is negative. So, even in the complex realm, the real part of these roots is negative. The roots are 'anchored' to the left of the imaginary axis.

Think about what we have just proven. Whether the roots are real (and negative) or complex (with a negative real part), the result is the same.

The roots of the equation $a{x}^2+bx+c=0$ with $a,b,c>0$ always have negative real parts. This is the elegance of mathematics—a single, unified truth emerging from two different scenarios.

Keep this intuition with you. When you see a quadratic equation in the future, don't just solve it; visualize it. You are not just calculating numbers; you are mapping the landscape of functions.