Problem 25 - Intermediate Algebra

Question

Solve. \(-\sqrt{2 x}+4=-6\)

Step-by-Step Solution

Answer

The solution is \( x = 50 \).

Step 1: Isolate the Square Root

To isolate the square root, first subtract 4 from both sides of the equation: \[ -\sqrt{2x} = -6 - 4 \]Simplify the right side:\[ -\sqrt{2x} = -10 \]

Step 2: Solve for the Square Root

Divide both sides by -1 to get:\[ \sqrt{2x} = 10 \]

Step 3: Square Both Sides

To eliminate the square root, square both sides:\[ (\sqrt{2x})^2 = 10^2 \] Simplify to:\[ 2x = 100 \]

Step 4: Solve for x

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

Key Concepts

Square Root

Understanding the square root is a vital part of tackling equations that involve them. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, because 3 times 3 equals 9. The square root symbol, \( \sqrt{} \), is used to denote this operation.

• Positive and negative numbers can both have square roots, but this typically refers to the positive (or principal) square root.
• In equations like \( -\sqrt{2x} + 4 = -6 \), we specifically work with the principal square root.
• Understanding how to manipulate and remove the square root is key to solving such problems.

By isolating and eliminating the square root, as shown in solving steps, you transform a complex equation into a simpler form.

Isolation Technique

The isolation technique is a method used to solve equations more easily by isolating one component of the equation on one side. For equations involving variables and numbers, this usually means getting the variable by itself.

• To begin with exercises like \( -\sqrt{2x} + 4 = -6 \), start by isolating the square root. You achieve this by subtracting 4 from both sides.
• This results in shifting all constant terms to one side, thereby focusing on the variable's side.
• The main goal is to make subsequent steps, such as squaring, more straightforward.

By using this technique, you pave the way towards simplifying the equation and allow easier access to solve for the variable.

Squaring Both Sides

Squaring both sides of an equation is a strategic move to eliminate square roots. This technique is especially employed when dealing with equations where square roots are isolated.

• For example, when you have \( \sqrt{2x} = 10 \), squaring each side removes the square root.
• This results in an equation where \( (\sqrt{2x})^2 = 10^2 \), simplifying to \( 2x = 100 \).
• It's important to note that squaring can sometimes introduce extraneous solutions, which need verifying against the original equation.

This method vastly simplifies the terms, making it easier to solve for the unknown variable by progressing to normal algebraic-solving strategies.

Simplifying Equations

Simplifying equations is the final step toward finding the solution. This step involves reducing the equation to its simplest form, making it manageable to solve.

• After squaring both sides, you often end up with an equation like \( 2x = 100 \), which is much simpler to solve than the original.
• Divide both sides by 2 to directly solve for \( x \).
• Simplifying corrects any potential errors made in earlier transformations, ensuring accuracy in the final answer.

The act of simplification not only ensures clarity in solving but also confirms that the steps engaged lead us to actual solutions, as opposed to false ones which might arise in earlier squared transformations.