Equations Reducible to Quadratic Equations
Big Idea: Many equations that don't look like quadratic equations can be transformed into quadratic form using a clever substitution. Once in quadratic form, we solve using familiar methods — then convert back to find the original variable.
📌 What Is a Quadratic Equation?
Recall that a standard quadratic equation has the form:
$$ ax^2 + bx + c = 0 $$
But what happens when we encounter equations like these?
- $x^4 - 5x^2 + 4 = 0$
- $x - 7\sqrt{x} + 12 = 0$
- $x^{2/3} + 5x^{1/3} + 6 = 0$
- $\left(\dfrac{x+1}{x}\right)^2 - 5\left(\dfrac{x+1}{x}\right) + 6 = 0$
These are NOT quadratic in $x$, but they ARE quadratic in some expression involving $x$. That's the key insight!