Question

(28) solve: $x^4 - 27x = 0$

Explanation:

Step1: Factor out the common term

We start with the equation \(x^{4}-27x = 0\). The common factor in both terms is \(x\), so we factor it out:
\(x(x^{3}-27)=0\)

Step2: Solve for \(x\) from the first factor

When \(x = 0\), the equation \(x(x^{3}-27)=0\) is satisfied. So \(x = 0\) is one solution.

Step3: Solve the cubic equation

For the second factor \(x^{3}-27 = 0\), we can rewrite \(27\) as \(3^{3}\), so the equation becomes \(x^{3}-3^{3}=0\). Using the difference of cubes formula \(a^{3}-b^{3}=(a - b)(a^{2}+ab + b^{2})\), where \(a=x\) and \(b = 3\), we get:
\((x - 3)(x^{2}+3x + 9)=0\)

Step4: Solve for \(x\) from the linear factor

Setting \(x - 3=0\), we find \(x=3\) is another solution.

Step5: Solve the quadratic equation

For the quadratic equation \(x^{2}+3x + 9=0\), we use the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), where \(a = 1\), \(b=3\), and \(c = 9\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(3)^{2}-4\times1\times9=9 - 36=- 27\)
Since the discriminant is negative, the solutions for this quadratic are complex numbers:
\(x=\frac{-3\pm\sqrt{-27}}{2}=\frac{-3\pm3i\sqrt{3}}{2}\)

Answer:

The solutions are \(x = 0\), \(x=3\), \(x=\frac{-3 + 3i\sqrt{3}}{2}\), and \(x=\frac{-3-3i\sqrt{3}}{2}\)