A radical equation is an equation in which the variable is inside a radical, most commonly a square root. Let's analyze each option to determine which one fits this definition.
1. Option 1: [tex]\( x + \sqrt{5} = 12 \)[/tex]
- This equation has a square root, but it does not involve the variable [tex]\( x \)[/tex]. The square root only applies to the constant [tex]\( 5 \)[/tex].
2. Option 2: [tex]\( x^2 = 16 \)[/tex]
- This equation is a quadratic equation, not a radical equation. The variable [tex]\( x \)[/tex] is raised to the power of 2, but it is not inside a radical.
3. Option 3: [tex]\( 3 + x \sqrt{7} = 13 \)[/tex]
- In this equation, [tex]\( \sqrt{7} \)[/tex] is a constant multiplied by [tex]\( x \)[/tex]. The variable [tex]\( x \)[/tex] is not inside the radical; the radical only applies to the constant [tex]\( 7 \)[/tex].
4. Option 4: [tex]\( 7 \sqrt{x} = 14 \)[/tex]
- This equation clearly contains a radical sign with the variable [tex]\( x \)[/tex] inside it. The variable [tex]\( x \)[/tex] is under a square root.
Based on this analysis, Option 4: [tex]\( 7 \sqrt{x} = 14 \)[/tex] is the correct answer because it is the only one that includes the variable [tex]\( x \)[/tex] inside a radical.
So, the correct option is:
4
