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: $\quad \sqrt{7 b+5}=-10$
Solution: $b=$ $\square$
Answer (2)
Square both sides of the equation: ( 7 b + 5 ) 2 = ( − 10 ) 2 , which simplifies to 7 b + 5 = 100 .
Solve for b : 7 b = 95 , so b = 7 95 .
Check the solution in the original equation: 7 ( 7 95 ) + 5 = 10 .
Since the square root cannot be negative, there is no real solution: D NE .
Explanation
Understanding the Problem We are given the equation 7 b + 5 = − 10 . We need to solve for b .
Squaring Both Sides First, we square both sides of the equation to eliminate the square root: ( 7 b + 5 ) 2 = ( − 10 ) 2 This simplifies to: 7 b + 5 = 100
Isolating the Term with b Next, we subtract 5 from both sides of the equation: 7 b = 100 − 5 7 b = 95
Solving for b Now, we divide both sides by 7 to solve for b : b = 7 95
Checking the Solution We must check our solution in the original equation to ensure it is valid. Substituting b = 7 95 into the original equation, we get:\begin{align*}\sqrt{7(\frac{95}{7})+5} &= \sqrt{95+5} &= \sqrt{100} &= 10\end{align*}However, the original equation is 7 b + 5 = − 10 . Since the square root of a real number is always non-negative, the equation 7 b + 5 = − 10 has no real solution.
Final Answer Therefore, there is no real solution to the equation.
Examples
Consider a scenario where you are modeling the speed of a wave using the equation 7 x + 5 = v , where x is a physical parameter and v is the wave speed. If you find that the wave speed v is negative (e.g., v = − 10 ), it indicates that the model is not valid for the given conditions, as the square root cannot be negative in real numbers. This implies that the parameter x cannot take a value that results in a negative wave speed, highlighting the limitations of the model.
The equation 7 b + 5 = − 10 has no real solution because the square root of a real number cannot be negative. Therefore, the solution is b = D NE .
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