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Chapter 8: Problem 26

Solve each equation. Check all solutions. $$\sqrt{x-5}-3=4$$

Short Answer

Expert verified
The solution to the equation \(\sqrt{x-5}-3=4\) is \(x = 54\), which is valid.

Step by step solution

01

Isolate the square root

First, the square root needs to be isolated on one side of the equation. This can be done by adding 3 to both sides of the equation, resulting in \(\sqrt{x-5}= 7\).
02

Eliminate the square root

To eliminate the square root, both sides of the equation can be squared, resulting in \(x-5 = 49\). Squaring a square root cancels it out, leaving just what鈥檚 under it.
03

Solve for x

To isolate x on one side of the equation, add 5 to both sides. This gives \(x = 54\).
04

Check the solution

Substitute x = 54 back into the original equation, \(\sqrt{x-5}-3=4\). This gives \(\sqrt{54-5}-3=4\), which simplifies to \(7鈭�3=4\). As this is a true statement, the solution x = 54 is valid.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Equations
An algebraic equation like \(\sqrt{x-5}-3=4\) involves variables and requires methods to solve for unknown values. Equations can come in various forms, and it's important to understand the symbols and operations involved.

Solving algebraic equations requires balancing both sides, an idea similar to a balance scale. If you add or subtract something on one side, the same must be done on the other to maintain equality.

This exercise involves operations like addition, subtraction, and dealing with a square root, making it slightly more challenging than basic algebraic equations.
Isolate Square Root
The first step to solve the equation \(\sqrt{x-5}-3=4\) is to isolate the square root. Isolating means getting the square root term on one side of the equation by itself, free from any constants or coefficients.

In this case, you add 3 to both sides, which balances the equation, leading to \(\sqrt{x-5} = 7\). It's like peeling away layers to reach what's inside.

Isolating terms is critical because it simplifies the process of eliminating operations, such as the square root, from the equation, leading to easier solutions.
Squaring Both Sides
Once the square root is alone, the next move is to square both sides of the equation \(\sqrt{x-5} = 7\). Squaring is multiplying a number by itself, and it helps get rid of the square root.

Here, squaring both sides yields \((\sqrt{x-5})^2 = 7^2\), simplifying to \(x - 5 = 49\). This step transforms the equation, making it easier to solve for the variable.

Remember, when squaring both sides of an equation, it鈥檚 essential to be mindful of the possibility of introducing extraneous solutions, as not all results from squaring operations are valid.
Verification of Solution
Verification is crucial to ensure that your solution satisfies the original equation. After solving and finding \(x = 54\), substitute it back into the original equation to check if it holds true.

Place \(x = 54\) into \(\sqrt{x-5}-3=4\), resulting in \(\sqrt{54-5}-3=4\). Simplifying further gives \(7 - 3 = 4\), which confirms the equality is maintained.

This not only confirms the accuracy of your calculations but also ensures no mistakes were made along the way. Always remember to verify to avoid errors and confirm that your solution makes sense contextually.

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Most popular questions from this chapter

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt{\frac{500}{81}}$$

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt{\frac{320 x^{4} y}{x y^{3}}}$$

Simplify. All variables in square root problems represent positive values. Assume no division by 0. $$(\sqrt{8 x y}+1)(\sqrt{8 x y}+1)$$

Simplify. Assume that all variables in the radicand of an even root represent positive values. Assume no division by 0. Express each answer with positive exponents only. $$\left(3^{2 / 3} 5^{1 / 3}\right)^{3}$$

Simplify. Assume that all variables in the radicand of an even root represent positive values. Assume no division by 0. Express each answer with positive exponents only. $$32^{3 / 5}$$
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