Problem 25 - Precalculus

Question

Solve the equation or inequality. $$\sqrt{2 x+1}=3+\sqrt{4-x}$$

Step-by-Step Solution

Answer

The solution is \(x = 4\).

Step 1: Isolate One Square Root

We start by isolating one of the square root expressions. The equation is \( \sqrt{2x+1} = 3 + \sqrt{4-x} \), and one approach is to keep \( \sqrt{2x+1} \) on one side and \( 3 + \sqrt{4-x} \) on the other.

Step 2: Eliminate Square Root by Squaring

Square both sides of the equation to eliminate the square root on the left side: \[(\sqrt{2x+1})^2 = (3 + \sqrt{4-x})^2 \]This simplifies to:\[ 2x+1 = 9 + 6\sqrt{4-x} + (4-x) \]Combine terms:\[ 2x + 1 = 13 - x + 6\sqrt{4-x} \]

Step 3: Simplify and Isolate Remaining Square Root

Rearrange the equation to get:\[2x + 1 - 13 + x = 6\sqrt{4-x} \]This simplifies to:\[3x - 12 = 6\sqrt{4-x} \]Divide everything by 6:\[x - 4 = \sqrt{4-x} \]

Step 4: Square Again to Remove Remaining Square Root

Square both sides again to eliminate the second square root:\[(x - 4)^2 = (\sqrt{4-x})^2 \]Simplified, this becomes:\[ (x - 4)^2 = 4 - x \]Expanding the left side:\[ x^2 - 8x + 16 = 4 - x \]

Step 5: Rearrange and Solve the Quadratic Equation

Rearrange to form a quadratic equation:\[x^2 - 8x + 16 - 4 + x = 0 \]This simplifies to:\[x^2 - 7x + 12 = 0 \]Factor the quadratic:\[(x - 3)(x - 4) = 0 \]Thus, the solutions are \(x = 3\) and \(x = 4\).

Step 6: Verify the Solutions

Substitute \(x = 3\) back into the original equation:\[ \sqrt{2(3)+1} = 3 + \sqrt{4-3} \]Calculates to \(\sqrt{7} = 3 + 1\), which is false. Thus, \(x = 3\) is not a solution.Now, substitute \(x = 4\):\[\sqrt{2(4)+1} = 3 + \sqrt{4-4} \]Calculates to \(\sqrt{9} = 3 + 0\), which is true.Thus, \(x = 4\) is the only valid solution.

Key Concepts

Quadratic Equations

Quadratic equations are equations of the second degree, typically in the form \(ax^2 + bx + c = 0\). These types of equations can have two solutions, one solution, or no real solution at all. When we solve a quadratic equation, our goal is to find the value of \(x\) that makes the equation true. Quadratic equations are often solved by factoring, using the quadratic formula, or completing the square.
In our radical equation \( \sqrt{2x+1} = 3 + \sqrt{4-x}\), squaring both sides and rearranging the terms result in a quadratic equation \(x^2 - 7x + 12 = 0\). This equation is then factored into \((x - 3)(x - 4) = 0\), giving potential solutions \(x = 3\) and \(x = 4\). The factoring method was used in this case because it is often a quick and efficient way to solve quadratics, provided the equation is factorable.

Isolation of Square Roots

Isolating square roots is a crucial step in solving equations that include radicals. The main goal is to have the square root term by itself on one side of the equation. This process simplifies the equation, making it easier to eliminate the radical by squaring.

• Begin with the equation \(\sqrt{2x+1} = 3 + \sqrt{4-x}\).
• Isolate one square root by arranging terms: keep \(\sqrt{2x+1}\) on one side and \(3 + \sqrt{4-x}\) on the other.
• After rearranging and simplifying, continue with \( x - 4 = \sqrt{4-x} \).

Through this isolation process, squaring becomes more straightforward, allowing the elimination of radicals and progression towards solving the equation.

Verification of Solutions

Verification is a vital step to ensure that the solutions derived for an equation are indeed valid. This is particularly important with radical equations, as squaring can introduce extraneous solutions—solutions that don't satisfy the original equation.First, substitute each potential solution back into the original equation:- **For \(x = 3\):** Substitute to find \(\sqrt{2(3)+1} = 3 + \sqrt{4-3}\). This simplifies to \(\sqrt{7} = 4\), which is false. So \(x = 3\) is not valid.- **For \(x = 4\):** Substitute to get \(\sqrt{2(4)+1} = 3 + \sqrt{4-4}\), which simplifies to \(3 = 3\), a true statement.
Thus, verification shows that \(x = 4\) is the only correct solution. Always double-check potential solutions, especially when dealing with square roots.

Simplification of Algebraic Expressions

Simplification involves combining like terms and performing arithmetic operations to make an expression or equation easier to work with. This is a frequent part of solving math problems, particularly when dealing with complex expressions.
In solving \(\sqrt{2x+1} = 3 + \sqrt{4-x}\), we move through several simplification steps:- After squaring both sides the first time, we expand \((3 + \sqrt{4-x})^2\) to get \(9 + 6\sqrt{4-x} + (4-x)\).- Combine like terms: \(2x + 1 = 13 - x + 6\sqrt{4-x}\).- Further simplification involves rearranging: \(3x - 12 = 6\sqrt{4-x}\).Simplifying correctly at each step avoids mistakes and helps in arriving at the right solution efficiently.