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Chapter 9: Problem 37

Solve equation by using the square root property. Simplify all radicals. \((x-3)^{2}=25\)

Short Answer

Expert verified
The solutions are \( x = 8 \) and \ (x = -2) \.

Step by step solution

01

- Isolate the Squared Term

Notice that the squared term \( (x-3)^2 \) is already isolated on one side of the equation. The equation \( (x-3)^2=25 \) is ready to be solved using the square root property.
02

- Apply the Square Root Property

To remove the square, apply the square root to both sides of the equation. Remember that taking the square root of both sides will give two possible solutions: \[ \sqrt{(x-3)^2} = \pm \sqrt{25} \]
03

- Simplify the Square Roots

Simplify both sides: \[ x-3 = \pm 5 \]
04

- Solve for x

Split the equation into two separate equations to solve for \( x \). \((x-3 = 5) \) and \((x-3 = -5)\). Then solve each: \[ x-3 = 5 \rightarrow x = 8 \] \[ x-3 = -5 \rightarrow x = -2 \]
05

- Final Answer

The solutions to the equation are \( x = 8 \) and \( x = -2 \). Both are correct and complete the solution.

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Key Concepts

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

square root property
The square root property is a useful tool for solving equations involving squared terms. It states that for any real number \(a\), if \(x^2 = a\), then \(x = \pm \sqrt{a}\). This means that taking the square root of both sides of an equation gives us two possible solutions.

Let's apply this to our exercise, \((x-3)^2 = 25\). The squared term is already isolated, so we can use the square root property immediately:
\[ \sqrt{(x-3)^2} = \pm \sqrt{25} \]
Simplifying the square roots gives us \(x-3 = \pm 5\). Note that we have two separate equations to solve: \(x-3 = 5\) and \(x-3 = -5\).
simplify radicals
Simplifying radicals is the process of breaking down a radical expression into its simplest form. In our exercise, we encounter the term \(\sqrt{25}\). To simplify this, we need to determine the number that, when squared, gives 25.

Here are the steps:
  • Identify the perfect square under the radical: 25.
  • Determine the square root of 25, which is 5.

Taking the square root of \(25\) gives us two possible values: \( \pm 5\). Therefore, \( \sqrt{25} = 5\) or \( -5\).

This simplification helps us make the equation easier to solve:
\[ x - 3 = \pm 5 \]
equation solving steps
Solving an equation involves a series of steps that simplify and unravel the expression to find the value of the variable. Here鈥檚 a breakdown of how we solved the equation \((x-3)^2 = 25\):
  • Step 1: Identify and isolate the squared term. In this case, the term \((x-3)^2\) is already isolated.
  • Step 2: Apply the square root property to both sides of the equation to eliminate the square: \[ \sqrt{(x-3)^2} = \pm \sqrt{25} \]
  • Step 3: Simplify the radicals: \(x-3 = \pm 5\).
  • Step 4: Solve for \(x\) by splitting into two separate equations:
    • \(x - 3 = 5\) → solve to get \(x = 8\).
    • \(x - 3 = -5\) → solve to get \(x = -2\).
  • Final Answer: The solutions are \(x = 8\) and \(x = -2\).
Remember to double-check your solutions by plugging them back into the original equation. This ensures your answers are correct.

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Most popular questions from this chapter

Solve each quadratic equation for complex solutions by the square root property, with \(k<0 .\) Write solutions in standard form. $$ (x-3)^{2}=-5 $$

Write each quotient in standard form. $$ \frac{-4+2 i}{1+i} $$

Use the quadratic formula to solve each equation. (a) Give solutions in exact form, and (b) use a calculator to give solutions correct to the nearest thousandth. $$ 2 x^{2}=5-2 x $$

Solve each quadratic equation for complex solutions by the quadratic formula. Write solutions in standard form. $$ 4 q^{2}-2 q+3=0 $$

Simplify all radicals, and combine like terms. Express fractions in lowest terms. \(\frac{4}{5}+\sqrt{\frac{48}{25}}\)
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