Problem 62 - Algebra for College Students

Question

Your friend makes an effort to solve the equation \(3+2 \sqrt{x}=x\) as follows: $$ \begin{array}{r} (3+2 \sqrt{x})^{2}=x^{2} \\ 9+12 \sqrt{x}+4 x=x^{2} \end{array} $$ At this step he stops and doesn't know how to proceed. What help would you give him?

Step-by-Step Solution

Answer

To help, isolate and eliminate the square root by squaring, solve for \(x\), and verify valid solutions.

Step 1: Understand the Existing Expansion

Look at the expansion performed: \[(3+2 \sqrt{x})^2 = x^2\] and understand how it arrives at \[9+12 \sqrt{x}+4x = x^2.\] Verify that the expansion follows \[(a+b)^2 = a^2 + 2ab + b^2.\] Thus here, \[a = 3, \quad b = 2 \sqrt{x}.\] Check that the friend's work is correct so far.

Step 2: Move Terms to Isolate Square Root

Subtract all terms from both sides of the equation to bring the terms involving \(\sqrt{x}\) on one side: \[12 \sqrt{x} = x^2 - 4x - 9.\] This isolates the expression containing the square root on the left side of the equation.

Step 3: Square Both Sides

To eliminate the square root on the left side, square both sides of the equation: \[(12 \sqrt{x})^2 = (x^2 - 4x - 9)^2.\] Simplify the left side: \[144x = (x^2 - 4x - 9)^2.\] Ensure correct expansion of both sides.

Step 4: Further Simplification and Solve

Focus simplifying the right side using the generic formula \[(a-b-c)^2 = a^2 - 2ab - 2ac + 2bc + b^2 + c^2.\]You will end with a polynomial equation. Solve using methods appropriate like factoring or the quadratic formula.

Step 5: Checking Solutions

Substitute potential solutions from the polynomial back into the original equation, \[3 + 2\sqrt{x} = x,\] to check if they hold valid. Discard any extraneous solutions introduced during the squaring steps.

Key Concepts

Quadratic Equation

When dealing with equations involving radicals, we often encounter what is known as a **quadratic equation**. A quadratic equation is a polynomial equation of degree 2, generally in the form \(ax^2 + bx + c = 0\). In solving our given problem, the equation \[ (3 + 2 \sqrt{x})^2 = x^2 \]transforms into \[ 9 + 12 \sqrt{x} + 4x = x^2. \]This indicates that after expanding and simplifying, we are led towards creating a quadratic equation as part of our problem-solving process. Quadratic equations can be solved using several methods, such as factoring, applying the quadratic formula, or completing the square. In our situation, we also had to take care of the square roots to simplify the problem into a quadratic equation.

Isolation of Terms

The strategy of **isolation of terms** is crucial when handling complex equations. The primary aim here is to rearrange the equation in such a way that each variable is isolated on different sides of the equation. In this exercise, we started with:\[ 9 + 12 \sqrt{x} + 4x = x^2. \]To isolate terms related to the square root, we needed to bring all other terms to one side of the equation, leading to:\[ 12 \sqrt{x} = x^2 - 4x - 9. \]By doing so, the term containing \( \sqrt{x} \) is isolated, and subsequently, this enables easier manipulation, notably when aiming to eliminate the square root, which is our next step.

Elimination of Square Roots

Eliminating square roots is a common step when solving radical equations, allowing the equation to become more solvable through algebraic means. Once terms are isolated as\[ 12 \sqrt{x} = x^2 - 4x - 9, \]the subsequent task is to eliminate the \( \sqrt{x} \). This is done by squaring both sides of the equation:\[ (12 \sqrt{x})^2 = (x^2 - 4x - 9)^2. \]Squaring both sides cancels out the square root, giving us:\[ 144x = (x^2 - 4x - 9)^2. \]This step transforms the problem into a quadratic equation. However, it is vital to note that squaring both sides might introduce extraneous solutions, so care must be taken in the final verification step.

Verification of Solutions

The **verification of solutions** is an essential final step to ensure that solutions derived from squaring the equation are valid. Once a square root is removed by squaring, it is possible to introduce solutions that do not satisfy the original equation. Thus, each solution must be checked back in the original formulation, which here is:\[ 3 + 2 \sqrt{x} = x. \]For example, if we solve and find potential solutions for \(x\), each should be substituted back into this equation to confirm validity. If a solution does not satisfy the initial equation, it should be labeled as extraneous and therefore discarded. This step guarantees that our final answers are genuine and accurate representations consistent with the starting problem.