Animated Solution for Mathematics - Quadratic Equations: If $b>a$, then the equation $(x-a)(x-b)-1=0$ has

A Parabolic Journey

Imagine you are standing on the x-axis, looking at the graph of a quadratic function. The equation $(x-a)(x-b)-1=0$ might look like a simple algebraic expression, but it is actually a story about a curve dancing across the coordinate plane.

Today, we are going to uncover where this curve crosses the x-axis without ever needing to solve for $x$ directly.

Let us define our function as $f(x)=(x-a)(x-b)-1$. We are given that $b>a$.

If we were to ignore the $-1$ for a moment, the roots of $(x-a)(x-b)=0$ would be exactly $a$ and $b$. But that $-1$ changes everything; it pulls the entire parabola downwards by one unit.

To understand where the new roots lie, we must test the function at these critical points, $a$ and $b$.

Let us substitute $x=a$ into our function:

Since $(a-a)=0$, the first term vanishes, leaving us with $f(a)=-1$. Because $-1<0$, we know that at $x=a$, the graph is below the x-axis.

Now, let us do the same for $x=b$:

Again, the term $(b-b)$ becomes zero, leaving us with $f(b)=-1$. Once again, the graph is below the x-axis at $x=b$.

We have discovered a vital clue: the parabola dips below the x-axis at both $a$ and $b$.

If we expand our function, we get $f(x)=x^2-(a+b)x+ab-1$. The coefficient of $x^2$ is $1$, which is positive.

This tells us that our parabola opens upwards, like a smile. This is the key to the entire puzzle.

Because it opens upwards, we know that as $x$ moves toward positive infinity ($x\to \infty$) or negative infinity ($x\to -\infty$), the function $f(x)$ must eventually shoot up toward positive infinity.

Now, let us connect the dots. We know that at $x=a$, the function is negative ($f(a)=-1$).

We also know that as $x$ goes to $-\infty$, the function is positive. By the Intermediate Value Theorem, if a continuous function goes from a positive value to a negative value, it must cross the x-axis.

Therefore, there must be a root somewhere in the interval $(-\infty ,a)$.

We apply the same logic to the right side. We know that at $x=b$, the function is negative ($f(b)=-1$).

We also know that as $x$ goes to $\infty$, the function is positive. To get from the negative dip at $b$ back up to positive infinity, the curve must cross the x-axis again.

Thus, there must be a second root in the interval $(b,\infty )$.

By simply analyzing the signs at $a$ and $b$ and understanding the shape of the parabola, we have successfully located both roots.

One root hides in the interval $(-\infty ,a)$, and the other resides in $(b,\infty )$. This is the elegance of mathematics—using the properties of functions to see the invisible.