Question

1. $x^2 + 8x = 0$

Explanation:

Step1: Factor out x

We can factor out the common term \( x \) from the left - hand side of the equation \( x^{2}+8x = 0 \). Using the distributive property \( ab + ac=a(b + c) \), where \( a=x \), \( b = x \) and \( c = 8 \), we get \( x(x + 8)=0 \).

Step2: Apply the zero - product property

The zero - product property states that if \( ab=0 \), then either \( a = 0 \) or \( b=0 \) (or both). So, for the equation \( x(x + 8)=0 \), we have two cases:
Case 1: \( x=0 \)
Case 2: \( x + 8=0 \). Solving \( x+8 = 0 \) for \( x \), we subtract 8 from both sides of the equation. So \( x=-8 \).

Answer:

\( x = 0 \) or \( x=-8 \)