Determine the Roots/Zeroes

Question

The equation x^4−5x^2+6=0 has four solutions. Place the four solutions in increasing order.

Are the solutions the square root of 3, -3, 2, and -2?

Answer 1

Hi, Meghan!

I believe your answer is CORRECT.

This type of equation is called a Biquadratic Equation. Web-searching this topic might give you many examples. As you probably know, the trick here is to rename x² as y, that is, y= x². Take a look:

We start with

x⁴−5x²+6=0.

First, rewrite the exponents to make it clearer:

(x²)²-5(x²)+6=0.

Then, replace x² with y.

y²-5y+6=0.

This is now a regular quadratic equation, that we can solve by factoring. Look for two numbers that add up to -5 and multiply to 6. By thinking a little bit, we find that the numbers are -2 and -3. Then we get

(y-2)(y-3)=0,

that is,

y=2 or y=3.

Finally, we need to remember that we said y=x², so we have

x²=2 or x²=3,

Taking the square root and noting that the answer could be either positive or negative, we have

x= ±√2 or x=±√3.

Good job!