Question

Solve each equation.

$x^4-5x^2+4=0$

Short answer

The solutions are-2, -1, 1and2.

Step-by-step solution

Step 1. Given Information.

Given, to solve the equation $x^4-5x^2+4=0$.

Step 2. Calculation.

Solving the equation:

$\begin{array}{l}{\left(x^2\right)}^2-5\left(x^2\right)+4=0\\ {\left(x^2\right)}^2-\left(x^2\right)-4\left(x^2\right)+4=0\left(\text{Writingitinthequadraticform}\right)\\ \left(x^2\right)\left(x^2-1\right)-4\left(x^2-1\right)=0\\ \left(x^2-1\right)\left(x^2-4\right)=0\left(\text{Factoringthetrinomial}\right)\\ \left(x^2-1^2\right)\left(x^2-2^2\right)=0\\ \left(x-1\right)\left(x+1\right)\left(x-2\right)\left(x+2\right)=0\left(\text{Factoringeachdifferenceofsquares}\right)\end{array}$

Using the Zero Product Property:

$\begin{array}{l}x-1=0\text{or}x+1=0\text{or}x-2=0\text{or}x+2=0\\ x=1\text{or}x=-1\text{or}x=2\text{or}x=-2\end{array}$

Step 3. Conclusion.

Hence, the solutions are -2, -1, 1 and 2.