Solve each equation. √(x+7)+3=√(x-4)

Verified step by step guidance

1. Start by isolating one of the square root expressions. For example, subtract 3 from both sides to get: \(\sqrt{{x+7}} = \sqrt{{x-4}} - 3\).
2. Next, to eliminate the square roots, square both sides of the equation. Remember to apply the formula for squaring a binomial on the right side: \(\left(\sqrt{{x-4}} - 3\right)^2\).
3. After squaring, simplify both sides. The left side becomes \(x+7\), and the right side expands to \((x-4) - 2 \cdot 3 \cdot \sqrt{{x-4}} + 9\).
4. Rearrange the equation to isolate the remaining square root term. This will involve moving all non-root terms to one side and keeping the root term on the other side.
5. Square both sides again to eliminate the remaining square root, then simplify and solve the resulting quadratic equation for \(x\). Finally, check your solutions in the original equation to avoid extraneous roots.

Key Concepts

Square Root Equations

Square root equations involve variables inside a radical sign. To solve them, isolate the square root on one side and then square both sides to eliminate the radical. This process may introduce extraneous solutions, so checking all solutions in the original equation is essential.

Domain Restrictions

The domain of a square root function includes only values that make the radicand (expression inside the root) non-negative. For the equation √(x+7) + 3 = √(x-4), both x+7 and x-4 must be greater than or equal to zero to ensure real solutions.

Checking for Extraneous Solutions

Squaring both sides of an equation can introduce solutions that do not satisfy the original equation. After solving, substitute each solution back into the original equation to verify its validity and discard any extraneous solutions.