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Chapter 10: Problem 10

Solve the given equation. $$\sqrt{x-4}=7$$

Short Answer

Expert verified
The solution is \(\textstyle x=53\).

Step by step solution

01

Understand the Equation

The equation given is \(\textstyle\sqrt{x-4}=7\). This equation means that the square root of \(\textstyle{x-4}\) equals 7.
02

Isolate the Radical

The radical is already isolated on one side: \(\textstyle\sqrt{x-4}=7\).
03

Square Both Sides

To eliminate the square root, square both sides of the equation: \(\textstyle(\sqrt{x-4})^2 = 7^2\). This simplifies to \(\textstyle{x-4 = 49}\).
04

Solve for x

Add 4 to both sides to solve for \(\textstyle{x}\): \(\textstyle{x-4 + 4 = 49 + 4\), which simplifies to \(\textstyle{x = 53}\).
05

Verify the Solution

To ensure the solution is correct, substitute \(\textstyle{x=53}\) back into the original equation: \(\textstyle\sqrt{53-4} = \sqrt{49} = 7\), which is a true statement.

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Key Concepts

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

Radical Equations
Radical equations are equations where the unknown variable appears under a radical sign, most commonly a square root. Solving these equations involves several specific steps. The main goal is to simplify and eliminate the radical sign to solve for the variable inside it.

In the context of the given exercise, the equation \[ \textstyle\sqrt{x-4}=7 \] is a classic example of a radical equation because the variable \ x \ is inside the square root. Before we can solve it, we need to deal with the radical component.
Isolation Techniques
Isolation techniques are key when solving equations, especially radical ones. Isolation involves manipulating the equation to get the variable or its radical expression by itself on one side.

In our problem, the radical is already isolated on one side of the equation \[ \textstyle\sqrt{x-4}=7 \] so no further isolation steps are needed. But generally, to isolate a radical, you might need to:
  • Move all other terms to the opposite side of the equation
  • Simplify any constants or coefficients outside of the radical
This simplifies the equation into a more manageable form, ready for the next step.
Verification of Solutions
Verification of solutions is a crucial final step in solving any equation, especially those involving radicals. This step ensures that the solution does indeed satisfy the original equation.

To verify, substitute the solution back into the original equation and confirm that both sides are equal. In our example:
  • Original Equation: \[ \textstyle\sqrt{x-4}=7 \]
  • Proposed Solution: \ x = 53 \
  • Substitute 53 into the original equation: \[ \textstyle\sqrt{53-4}=7 \]
  • This simplifies to: \[ \textstyle\sqrt{49}=7 \]
  • Since \ 7 = 7 \, our solution is confirmed.

Make it a habit to verify your solutions to avoid errors and confirm the accuracy of your results.

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Most popular questions from this chapter

Verify that \(2+\sqrt{10}\) is a solution to the equation \(x^{2}-4 x-6=0\).

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\sqrt{20 y^{3}}-\sqrt{45 y^{3}}$$

Factor the given expression as completely as possible. $$w^{3}-8 w^{2}+16 w$$

Use a calculator to give decimal approximations rounded to one, two, and three places. $$\sqrt{23}$$

Reduce to lowest terms. $$\frac{4+\sqrt{8}}{6}$$
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