A Very Tricky Quadratic Equation

What to do if we have an absolute sign?

Quadratic equations are of the form ax² + bx + c = 0, where a, b and c are real values. In school, we learnt to solve such equations by factoring or using the quadratic formula.

What happens if instead we have |x|? Recall that |-3| = 3 for example.

Here’s a hint: a substitution might help …

I recommend you pause the article, grab your pen and paper, and give this a go. When you are ready, keep reading for the solution! ✒️

Solution

Let us take a leap of faith by letting y = |x| so our equation turns into

How would you now solve this equation for y?

We can factorize it into two factors like below

We can check that (y − 1)(y + 5) = y² + 5y − y − 5 = y² + 4y − 5.

Now that means each of the factors equals zero.

Do you see where we are going with this?

Let us look at our initial substitution y = |x|.

Does it make sense that for real values of x, we can have |x| = 1, what about |x| = -5?

By definition |x| must be greater than or equal to zero for all real values of x, this means we can omit |x| = −5.

So we are left with

So only x = 1 and x = -1 can satisfy the equation.

How amazing 👧

What was your thought process this time? Comment down below, I am eager to know :) 💞

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