What is the solution of $\sqrt{1-3 x}=x+3$?
A. $x=-8$ or $x=-1$
B. $x=-8$
C. $x=-1$
D. no solution
Answer (2)
The solution to the equation 1 − 3 x = x + 3 is x = − 1 , which is a valid solution after squaring both sides and checking for extraneous solutions. The other potential solution, x = − 8 , is extraneous and does not satisfy the original equation.
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Square both sides of the equation 1 − 3 x = x + 3 to get 1 − 3 x = ( x + 3 ) 2 .
Simplify and rearrange the equation into a quadratic equation: x 2 + 9 x + 8 = 0 .
Solve the quadratic equation by factoring: ( x + 1 ) ( x + 8 ) = 0 , which gives x = − 1 or x = − 8 .
Check the solutions in the original equation. x = − 1 is a valid solution, but x = − 8 is an extraneous solution. The final answer is − 1 .
Explanation
Problem Analysis We are given the equation 1 − 3 x = x + 3 and asked to find its solution. We will solve this equation by squaring both sides and then checking for extraneous solutions.
Squaring Both Sides First, we square both sides of the equation to eliminate the square root: ( 1 − 3 x ) 2 = ( x + 3 ) 2 This simplifies to: 1 − 3 x = x 2 + 6 x + 9
Rearranging into Quadratic Form Next, we rearrange the equation into a quadratic equation by moving all terms to one side: x 2 + 6 x + 9 − 1 + 3 x = 0 x 2 + 9 x + 8 = 0
Solving the Quadratic Equation Now, we solve the quadratic equation by factoring:We look for two numbers that multiply to 8 and add to 9. These numbers are 1 and 8. So, we can factor the quadratic as follows: ( x + 1 ) ( x + 8 ) = 0 This gives us two possible solutions: x + 1 = 0 ⇒ x = − 1 x + 8 = 0 ⇒ x = − 8
Checking for Extraneous Solutions We must check these solutions in the original equation to eliminate extraneous solutions.For x = − 1 : 1 − 3 ( − 1 ) = − 1 + 3 1 + 3 = 2 4 = 2 2 = 2 So, x = − 1 is a valid solution.For x = − 8 : 1 − 3 ( − 8 ) = − 8 + 3 1 + 24 = − 5 25 = − 5 5 = − 5 This is false, so x = − 8 is an extraneous solution and not a valid solution.
Final Answer Therefore, the only solution to the equation 1 − 3 x = x + 3 is x = − 1 .
Examples
When designing a bridge or any structure involving square roots in calculations, it's crucial to verify the solutions because extraneous solutions can lead to incorrect measurements and potentially catastrophic outcomes. For instance, if the length of a support beam is determined by an equation involving a square root, and an extraneous solution is used, the beam might be too short or too long, compromising the structural integrity.
