Problem 127 - Elementary and Intermediate Algebra

Question

Solve each equation. Write all proposed solutions. Cross out those that are extraneous. $$ \sqrt{x+2}+\sqrt{2 x}=\sqrt{18-x} $$

Step-by-Step Solution

Answer

The valid solution is \(x = 2\); the extraneous solution is \(x = -16\).

Step 1: Isolate Radicals

First, we need to rearrange the equation so that similar terms are isolated. Start by noting that moving terms isn't straightforward due to the radicals. But we know that squaring both sides of an equation typically helps eliminate the radicals.

Step 2: Square Both Sides

Square the entire equation to remove the radicals. That means we apply \[ (\sqrt{x+2}+\sqrt{2x})^2 = (\sqrt{18-x})^2 \]The right side simplifies to:\[18-x \]

Step 3: Expand and Simplify

After squaring, the left-hand side is: \[ (\sqrt{x+2} + \sqrt{2x})^2 = (\sqrt{x+2})^2 + 2 \cdot \sqrt{x+2} \cdot \sqrt{2x} + (\sqrt{2x})^2 \]Simplify this as:\[ (x+2) + 2\sqrt{2x(x+2)} + 2x \]So our equation becomes:\[ x+2 + 2\sqrt{2x(x+2)} + 2x = 18 - x \]

Step 4: Combine Like Terms

Combine like terms on the left:\[ 3x + 2 + 2\sqrt{2x(x+2)} = 18 - x \]To further simplify, move all terms except the radical term to the right-hand side:\[2\sqrt{2x(x+2)} = 16 - 4x \]

Step 5: Isolate and Square Again

Divide everything by 2:\[\sqrt{2x(x+2)} = 8 - 2x\]To remove the square root, square both sides again:\[2x(x+2) = (8 - 2x)^2\]Simplify the right-hand side:\[(8 - 2x)^2 = 64 - 32x + 4x^2\]

Step 6: Solve the Quadratic Equation

Now we have:\[2x(x+2) = 64 - 32x + 4x^2\]Which simplifies to:\[2x^2 + 4x = 64 - 32x + 4x^2\].Rearrange terms:\[0 = 4x^2 + 32x - 2x^2 - 4x - 64 \]Which further simplifies to the quadratic equation:\[ 2x^2 + 28x - 64 = 0\] Divide everything by 2:\[x^2 + 14x - 32 = 0\]

Step 7: Factor or Use Quadratic Formula

Factor or use the quadratic formula. Check first if it's factorable. If not, use the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), with \(a = 1\), \(b = 14\), \(c = -32\). \[ b^2 - 4ac = 196 + 128 = 324 \], which is a perfect square. Thus:\[ x = \frac{-14 \pm \sqrt{324}}{2} \],\[ x = \frac{-14 \pm 18}{2} \].

Step 8: Solve for x

Evaluate potential solutions:\[ x = \frac{4}{2} = 2 \] \[ x = \frac{-32}{2} = -16 \] We get potential solutions \(x = 2\) and \(x = -16\).

Step 9: Check for Extraneous Solutions

Plug each solution back into the original equation to check:For \(x = 2\): \[\sqrt{2+2} + \sqrt{4} = \sqrt{18-2}\]\[2 + 2 = 4\]\[4 = 4 \] It holds true.For \(x = -16\):\(\sqrt{-16+2} + \sqrt{-32} = \sqrt{18+16}\), The square root of a negative number is not real, hence the solution \(x = -16\) is extraneous.

Step 10: Conclusion of Solution

The only valid solution is \(x = 2\). The extraneous solution \(x = -16\) does not satisfy the original equation under real numbers.

Key Concepts

Radical Expressions

Radical expressions, often seen in mathematical equations, involve roots of numbers or variables. A common radical is the square root, represented by the symbol \( \sqrt{} \). Solving equations with radical expressions entails manipulating these roots to simplify the equation or isolate variables.

When dealing with radical expressions, remember these key points:

• To "undo" a square root, you square the expression. For example, squaring \( \sqrt{x+2} \) results in \( x+2 \).
• It's crucial to apply operations to both sides of the equation to maintain equality. This includes squaring or adding equal terms.

Understanding radical expressions is essential because they can appear in different steps of solving equations. Whether they emerge from roots in algebra or simplifying expressions, radicals require careful handling to avoid mistakes.

Extraneous Solutions

Extraneous solutions occur when solving an equation produces results that don't fit back into the original equation. These are often introduced when both sides of an equation are squared to eliminate radical expressions.

Squaring an equation can sometimes "create" solutions that aren't valid in the context of the initial problem.
When you finish solving a quadratic equation, you might find more potential solutions than expected. To identify extraneous solutions:

• Substitute each potential solution back into the original equation.
• Check for realistic answers, especially considering the constraints of the original equation, such as domains where the values must be real numbers.

For example, a negative solution might seem plausible until you substitute it back and identify it's not acceptable. Always perform this check since excluding extraneous solutions prevents incorrect conclusions.

Solving Equations

The process of solving equations, especially those containing radicals or quadratic forms, is a key skill in mathematics. It involves systematically transforming the equation to find the value(s) of "x" or whatever variable is involved.

Steps often include:

• Isolating variables to one side of the equation. This might mean moving all terms involving "x" to one side.
• Simplifying expressions where possible, such as combining like terms.
• Using algebraic techniques like factoring, use of formulas such as the quadratic formula, or methods like completing the square.
• Checking your final answers by back-substituting them into the original equation to verify their validity.

If an equation involves radical expressions and extraneous solutions, particular attention is needed to ensure each solution is valid. Properly solving equations not only includes finding solutions but ensuring they genuinely satisfy the given problem.