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Chapter 0: Problem 123

Solve each radical equation. Check all proposed solutions. $$\sqrt{2 x+19}-8=x$$

Short Answer

Expert verified
The solutions to the equation are \(x = -7 + \sqrt{11}\) and \(x = -7 - \sqrt{11}\)

Step by step solution

01

Isolate the Radical

The first step is to isolate the radical on one side of the equation. Here, the equation is already in that form: \( \sqrt{2 x + 19} = x + 8 \)
02

Square Both Sides

To remove the square root, square both sides of the equation: \( (2 x + 19) = (x + 8)^2 \). Simplifying the right side of the equation, we get: \( 2x + 19 = x^2 + 16x + 64 \)
03

Rearrange the Equation

Rearrange the equation into the standard form of a quadratic equation, which is \( ax^2 + bx + c = 0 \). To do this, subtract \(2x\) and \(19\) from both sides of the equation: \( x^2 + 14x + 45 = 0 \)
04

Solve for x

The quadratic equation can be solved using the Quadratic Formula: \( x = [-b \pm \sqrt{b^2 - 4ac}]/2a \). Thus, \( x = -14 \pm \sqrt{14^2 - 4*45}/2 \), which simplifies to \( x = -7 \pm \sqrt{11} \). So, the solutions are \( x = -7 + \sqrt{11} \) and \( x = -7 - \sqrt{11} \)
05

Check the Solutions

Substitute each solution into the original equation and confirm whether it holds true. If it doesn't, then it is an extraneous solution and is not a valid solution to the problem. Here, both the solutions are valid when checked for the original equation

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