Solve each equation. See Examples 4–6. √(x+7)+3=√(x-4)

Verified step by step guidance

1. Start by isolating one of the square root terms. Subtract 3 from both sides to get $\sqrt{x+7}=\sqrt{x-4}-3$.
2. Square both sides of the equation to eliminate the square roots. This gives $x+7=(\sqrt{x-4}-3{)}^2$.
3. Expand the right side of the equation: $(\sqrt{x-4}-3{)}^2=(\sqrt{x-4}{)}^2-2\cdot 3\cdot \sqrt{x-4}+3^2$.
4. Simplify the expanded equation: $x+7=x-4-6\sqrt{x-4}+9$.
5. Combine like terms and solve for $\sqrt{x-4}$ to continue simplifying the equation.

Key Concepts

Square Roots

Square roots are mathematical expressions that represent a number which, when multiplied by itself, gives the original number. In the context of the equation, understanding how to manipulate square roots is essential for isolating variables and solving the equation effectively.

Isolating Variables

Isolating variables involves rearranging an equation to get the variable of interest on one side. This is a fundamental technique in algebra that allows for easier solving of equations, particularly when dealing with square roots and other operations.

Squaring Both Sides

Squaring both sides of an equation is a method used to eliminate square roots. By squaring both sides, you can transform the equation into a polynomial form, which is often easier to solve. However, this process can introduce extraneous solutions that must be checked against the original equation.