Problem 17 - Algebra 1

Question

Solve the equation. Check for extraneous solutions. $$\sqrt{x}+5=0$$

Step-by-Step Solution

Answer

The equation has no solution as the obtained solution is extraneous.

Step 1: Isolate the square root

First, to isolate the square root on one side of the equation, subtract 5 from both sides of the equation. This gives: \( \sqrt{x} = -5 \)

Step 2: Square both sides of the equation

Secondly, to eliminate the square root, square both sides of the equation: \( (\sqrt{x})^2 = (-5)^2 \), which simplifies to: \( x = 25 \)

Step 3: Check for extraneous solutions

Thirdly, substitute \( x = 25 \) into the original equation: \( \sqrt{25} + 5 = 0 \). Evaluating the square root and sum gives \( 5 + 5 = 10 \), not zero. Therefore, \( x = 25 \) is an extraneous solution and it doesn't satisfy the original equation.

Key Concepts

Extraneous Solutions

When solving equations that involve square roots, extraneous solutions often emerge. These are answers that fit the modified equation but not the original one.

For example, when you solve the equation \(\sqrt{x}+5=0\) by squaring both sides, you may introduce a solution that doesn't actually work when plugged back into the original equation. This is because squaring is not a reversible operation over all real numbers—when you square a negative number, you get a positive result, which loses the original sign information.

Therefore, it's crucial to always check your solutions by substituting them back into the original equation. This step verifies whether the solutions genuinely make the equation true. In our case, plugging \(x=25\) back into the original equation resulted in 10, not 0, which indicates that 25 is an extraneous solution.

Isolating the Square Root

To isolate the square root in an equation, you should first move all other terms to the opposite side of the equation. This simplification allows you to focus on the square root itself.

In our exercise, \(\sqrt{x}+5=0\), you move the +5 across the equals sign to isolate the square root, resulting in \(\sqrt{x} = -5\). It's a crucial step because it sets up the equation for the next move, which is to square both sides. Isolating the square root is like giving it its own space to work, without any extra baggage that can complicate things when you take further steps to solve.

Square Both Sides of the Equation

After isolating the square root, the next step is to square both sides of the equation. This is done to eliminate the square root and work with a simplified, algebraic expression.

Squaring the isolated square root, as you did with \(\sqrt{x}\), transforms it to \(x\). When you square the other side, you have to be mindful to square the entire side, not just the number closest to the equal sign. For instance, \(\left(-5\right)^{2} = 25\), not -25. It's a common mistake to forget that negative numbers also need to be squared as wholes. This process brings you closer to finding the solution, though you'll need to check for extraneous solutions as squaring both sides can introduce answers that don't work in the original equation.