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: $\sqrt{8 w-6}-4=0$
Solution: $w =$
Answer (2)
Isolate the square root term: 8 w − 6 = 4 .
Square both sides: 8 w − 6 = 16 .
Solve for w : 8 w = 22 .
Simplify the fraction: w = 4 11 .
The solution is 4 11 .
Explanation
Problem Setup We are given the equation 8 w − 6 − 4 = 0 and we want to solve for w .
Isolating the Square Root First, we isolate the square root term by adding 4 to both sides of the equation: 8 w − 6 = 4
Squaring Both Sides Next, we square both sides of the equation to eliminate the square root: ( 8 w − 6 ) 2 = 4 2 8 w − 6 = 16
Adding 6 to Both Sides Now, we solve for w . Add 6 to both sides: 8 w = 16 + 6 8 w = 22
Dividing by 8 Divide both sides by 8: w = 8 22
Simplifying the Fraction Finally, we simplify the fraction: w = 4 11
Checking the Solution We should check our solution to make sure it is valid. Substituting w = 4 11 back into the original equation: 8 ( 4 11 ) − 6 − 4 = 22 − 6 − 4 = 16 − 4 = 4 − 4 = 0 The solution is valid.
Final Answer Therefore, the solution to the equation 8 w − 6 − 4 = 0 is w = 4 11 .
Examples
Imagine you are designing a square garden and need to determine the length of each side. If the area inside the garden can be modeled by the equation 4 x − 3 − 5 = 0 , where x represents the length of each side, solving this equation will give you the required length. By isolating the square root, squaring both sides, and solving for x , you find the side length that satisfies the area requirement. This application demonstrates how solving equations with square roots can be used in practical design and measurement problems.
To solve the equation 8 w − 6 − 4 = 0 , we isolate the square root and square both sides to find w = 4 11 . We verified our solution by substituting it back into the original equation. Therefore, the solution is w = 4 11 .
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