Solve the equation, writing the solution as a reduced fraction or as an integer. If there is no real solution, enter "DNE." If there are multiple solutions, separate the solutions with a comma.
Solve: $\quad \sqrt{-7 m-8}-5=0$
Solution: $m=$ $\square$
Answer (1)
Isolate the square root: − 7 m − 8 = 5 .
Square both sides: − 7 m − 8 = 25 .
Solve for m : m = − 7 33 .
Check for extraneous solutions and confirm the answer: − 7 33 .
Explanation
Understanding the Problem We are given the equation − 7 m − 8 − 5 = 0 . Our goal is to solve for m , remembering to check for extraneous solutions since we are dealing with a square root.
Isolating the Square Root First, we isolate the square root term by adding 5 to both sides of the equation: − 7 m − 8 = 5
Squaring Both Sides Next, we square both sides of the equation to eliminate the square root: ( − 7 m − 8 ) 2 = 5 2 − 7 m − 8 = 25
Solving for m Now, we solve for m . Add 8 to both sides: − 7 m = 25 + 8 − 7 m = 33 Divide both sides by -7: m = − 7 33
Checking for Extraneous Solutions We must check for extraneous solutions by substituting m = − 7 33 back into the original equation: − 7 ( − 7 33 ) − 8 − 5 = 33 − 8 − 5 = 25 − 5 = 5 − 5 = 0 Since the solution satisfies the original equation, m = − 7 33 is the solution.
Final Answer Therefore, the solution to the equation − 7 m − 8 − 5 = 0 is m = − 7 33 .
Examples
Imagine you're designing a security system that relies on a sensor. The equation you solved is similar to one you might encounter when calibrating the sensor's sensitivity. The value of 'm' could represent a critical threshold that needs to be set correctly to avoid false alarms or missed detections. Solving such equations accurately ensures the system functions reliably, protecting valuable assets or providing timely warnings.
