What is the solution of [tex]$\sqrt{x+2}-15=-3$[/tex]?
A. [tex]$x=142$[/tex]
B. [tex]$x=232$[/tex]
C. [tex]$x=322$[/tex]
D. no solution
Answer (2)
The solution to the equation x + 2 − 15 = − 3 is x = 142 , which is option A.
;
Isolate the square root: x + 2 = 12 .
Square both sides: x + 2 = 144 .
Solve for x : x = 142 .
Verify the solution: 142 + 2 − 15 = − 3 , so the solution is 142 .
Explanation
Understanding the Problem We are given the equation x + 2 − 15 = − 3 . Our goal is to find the value of x that satisfies this equation. We will isolate the square root, square both sides, and then solve for x . Finally, we will check our solution to make sure it is valid.
Isolating the Square Root First, we isolate the square root term by adding 15 to both sides of the equation: x + 2 − 15 + 15 = − 3 + 15
Simplified Equation This simplifies to: x + 2 = 12
Squaring Both Sides Next, we square both sides of the equation to eliminate the square root: ( x + 2 ) 2 = 1 2 2
Equation without Square Root This simplifies to: x + 2 = 144
Solving for x Now, we solve for x by subtracting 2 from both sides: x + 2 − 2 = 144 − 2
Solution for x This gives us: x = 142
Checking the Solution Finally, we check if our solution is valid by substituting it back into the original equation: 142 + 2 − 15 = 144 − 15 = 12 − 15 = − 3
Valid Solution Since the equation holds true, our solution is valid.
Examples
Imagine you are designing a bridge and need to calculate the length of a support cable. The equation x + 2 − 15 = − 3 is a simplified version of the kind of equations engineers use to determine the tension and length of cables under load. Solving such equations ensures the bridge is safe and stable. Understanding how to manipulate and solve equations with square roots is crucial for ensuring structural integrity in engineering projects.
