Problem 103 - Elementary and Intermediate Algebra

Question

a. \(\sqrt{2 x}-10=0\) b. \(\sqrt{2 x}+10=0\)

Step-by-Step Solution

Answer

a. \(x = 50\) b. No real solutions.

Step 1: Isolate the Square Root

For equation (a) \( \sqrt{2x} - 10 = 0 \), add 10 to both sides to isolate the square root: \(\sqrt{2x} = 10\).

Step 2: Square Both Sides

Square both sides of the equation to eliminate the square root: \((\sqrt{2x})^2 = 10^2\) which simplifies to \(2x = 100\).

Step 3: Solve for x (Equation a)

Divide both sides by 2 to solve for \(x\): \(x = \frac{100}{2} = 50\).

Step 4: Isolate the Square Root

For equation (b) \(\sqrt{2x} + 10 = 0\), subtract 10 from both sides to get \(\sqrt{2x} = -10\).

Step 5: Determine Validity (Equation b)

Recognize that the square root of a number cannot be negative, thus \(\sqrt{2x} = -10\) has no real solutions because \(\sqrt{2x}\) must be greater than or equal to zero.

Key Concepts

Isolate the Square Root

When solving equations involving square roots, the first crucial step is to isolate the square root expression. Think of it as peeling the outer layer to reach the core part of the equation. For example, in the equation \(\sqrt{2x} - 10 = 0\), you need to move any numbers or terms that are outside of the square root to the other side of the equation. This will give you a clearer path to finding the value of \(x\).

• Add or subtract terms on both sides of the equation to leave the square root all by itself.
• In our example, adding 10 to both sides isolates the square root: \(\sqrt{2x} = 10\).

Once isolated, you can more easily perform further operations to solve for the variable inside the square root.

Squaring Both Sides

After isolating the square root, the next step is to eliminate it by squaring both sides of the equation. This step is like unlocking a box to see what's inside. When you square both sides, you effectively remove the square root, leaving you with a simpler algebraic expression.

• For instance, with \(\sqrt{2x} = 10\), squaring gives \((\sqrt{2x})^2 = 10^2\).
• This transforms to \(2x = 100\), which is much easier to solve.

Just remember, squaring both sides is legally binding in mathematics; whatever you do to one side, you must do to the other to maintain the equation's balance.

Recognizing Valid Solutions

Not all results from squaring both sides are valid. One common pitfall is recognizing when a solution might not actually satisfy the original equation, especially with square roots.

• Remember: a square root expression cannot equal a negative number.
• In the equation \(\sqrt{2x} = -10\), simply by looking, you can see something's off because square roots are always zero or positive.

Checking your solutions by substituting them back into the original equation can help confirm their validity. This step ensures you don't get sidetracked by extraneous solutions that don't make sense in context.

Real Solutions

When solving equations, particularly those with square roots, it is essential to determine which solutions are "real" and which don't logically fit. A 'real solution' means that it makes sense within the constraints of real numbers, without imaginary or extraneous aspects.

• For equations like \(\sqrt{2x} = 10\), solving gives \(x = 50\), a real and valid solution.
• However, for \(\sqrt{2x} = -10\), there simply are no real solutions, because the equation demands the impossible—a square root that isn't non-negative.

Real solutions tell us whether practical answers exist and help guide decisions in both math problems and real life scenarios.