Equations Reducible to Quadratic Form (Part 1)


Hello. In this lesson (and the following ones), I’ll talk about equations that are not quadratic equations, but are reducible or convertible to a quadratic equation.

These equations, that are otherwise difficult to solve, can be easily solved if we convert them to a quadratic form.

Let’s start with an easy example: x4 – 10x2 + 9 = 0.

This almost looks like a quadratic equation. It has a special name too – biquadratic equation. That is, an equation of the form ax4 + bx2 + c = 0. But let’s not get distracted.

Do you know how to proceed here? I’m sure you’d have figured it out by now.

One of the ways to begin is factorization.

The equation can be re-written as x4 – x2 – 9x2 + 9 = 0.

Next, we get x2(x2 – 1) – 9(x2 – 1) = 0, which finally gives us (x2 – 1)(x2 – 9) = 0.

Now, equating both factors to zero gives x2 = 1 and x2 = 9.

From these two equations, we get four solutions: 1, –1, 3, and –3.

But this method is not what I’m interested in. By ‘reducible’ to quadratic, I meant converting the equation to a quadratic one by making a (clever) substitution. Let me show you how.

Consider the original equation again: x4 – 10x2 + 9 = 0.

If we substitute x2 = y, the equation will look like y2 – 10y + 9 = 0. A quadratic equation!

We’ve now entered familiar territory. Using any of the methods in the previous lesson, we get y = 1 or 9.

But the story doesn’t end here. We were supposed to find the values of x, and not y.

So, we’ll substitute the value of y back in terms of x.

We get x2 = 1 or 9, which gives us the values of x as 1, –1, 3, and –3. Fancy, right?

 

To summarize, we’ll first make a substitution that converts the given equation to a quadratic one.

Then, we’ll solve the quadratic equation to find the value of the new variable (say y).

Finally, we’ll substitute the value of y back in terms of x to find the values of x.

But all equations aren’t as nice as biquadratic equations.

Substitutions will get trickier than x2 = y, and they’ll of course depend on the original equation’s form.

I’ll cover a few different reducible forms in the next couple of lessons. See you in next lesson.