In Exercises 1–38, solve each radical equation. _____ x = √3x + 7 - 3

Verified step by step guidance

1. Start by isolating the square root term on one side of the equation. Add 3 to both sides to get: $\sqrt{3x+7}=x+3$.
2. Square both sides of the equation to eliminate the square root. This gives: $3x+7=(x+3{)}^2$.
3. Expand the right side of the equation: $(x+3{)}^2=x^2+6x+9$.
4. Rearrange the equation to set it to zero: $3x+7=x^2+6x+9$ becomes $0=x^2+3x+2$.
5. Factor the quadratic equation: $x^2+3x+2=(x+1)(x+2)=0$.

Key Concepts

Radical Equations

Radical equations are equations in which a variable is contained within a radical (square root, cube root, etc.). To solve these equations, one typically isolates the radical on one side and then squares both sides to eliminate the radical. This process may introduce extraneous solutions, so it's important to check all potential solutions in the original equation.

Isolating the Variable

Isolating the variable involves rearranging the equation to get the variable alone on one side. This is a crucial step in solving equations, as it simplifies the problem and allows for easier manipulation. In the context of radical equations, isolating the radical before squaring both sides is essential to avoid complications.

Extraneous Solutions

Extraneous solutions are solutions that emerge from the process of solving an equation but do not satisfy the original equation. This often occurs when squaring both sides of a radical equation, as it can introduce solutions that are not valid. Therefore, it is critical to substitute any found solutions back into the original equation to verify their validity.