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Chapter 1: Problem 21

Solve equation by the square root property. $$ (x+2)^{2}=25 $$

Short Answer

Expert verified
The solutions for the equation \((x+2)^{2}=25\) are \(x = 3\) and \(x = -7\).

Step by step solution

01

Isolate the Square Term

The given equation is \((x+2)^{2}=25\). In this equation, \((x+2)^{2}\) is already isolated.
02

Apply the Square Root Property

The square root property states that given \(x^{2} = a\), \(x\) is either \(\sqrt{a}\) or \(-\sqrt{a}\). So, apply the square root to both sides of the equation: \(x + 2 = \sqrt{25}\) or \(x + 2 = -\sqrt{25}\).
03

Evaluate the Square Root

Evaluate the square root of 25 which is 5. This gives \(x + 2 = 5\) or \(x + 2 = -5\).
04

Solve for x

Subtract 2 from both sides of each equation to solve for \(x\). This provides two solutions: \(x = 5 - 2 = 3\) and \(x = -5 - 2 = -7\).

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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 powerful concept in mathematics, particularly when solving quadratic equations. It simplifies equations by taking the square root of both sides. When you have an equation in the form \(x^2 = a\), you can apply the square root property to find the value of \(x\). This allows \(x\) to be \(\sqrt{a}\) or \(-\sqrt{a}\).
This means once you know the value of \(a\), you can find two possible values for \(x\). When dealing with equations like \((x + 2)^2 = 25\), applying this property helps us solve it efficiently.
Isolating Terms
Isolating terms is an essential first step in solving equations involving the square root property. By isolating terms, we mean making sure that the squared portion of the equation stands alone on one side. This is crucial as it prepares the equation for applying the square root property.
In the given equation \((x+2)^2=25\), the term \((x+2)^2\) is already isolated, which makes our work easier. If a quadratic term isn't isolated, you would need to manipulate the equation, possibly by adding or subtracting terms from both sides, until you achieve this isolation.
Evaluating Square Roots
Evaluating square roots is a step where you determine the actual number value of a square root. Once you have isolated the square term and applied the square root property, the next step is to evaluate the square root itself.
For instance, in \((x+2)^2 = 25\), once isolated and square root applied, you get \(x+2 = \pm\sqrt{25}\). Evaluating \(\sqrt{25}\) gives us 5. Hence, \(x + 2\) equals both 5 and -5 at different occurrences.
  • Positive root: \(x + 2 = 5\)
  • Negative root: \(x + 2 = -5\)
Two Solutions
Quadratic equations often yield two solutions due to the nature of the square roots. After applying the square root property and evaluating the square roots, you'll typically have two scenarios to solve. In our ongoing example, \(x + 2 = 5\) and \(x + 2 = -5\). These equations offer different paths that lead to two correct solutions.
Solve each equation separately by isolating \(x\):
  • From \(x + 2 = 5\), subtract 2 to find \(x = 3\).
  • From \(x + 2 = -5\), subtract 2 to find \(x = -7\).
Hence, the equation has two solutions: \(x = 3\) and \(x = -7\). These solutions show the symmetry of solutions in quadratic equations.

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Most popular questions from this chapter

In a round-robin chess tournament, each player is paired with every other player once. The formula $$ N=\frac{x^{2}-x}{2} $$ models the number of chess games, \(N,\) that must be played in a round-robin tournament with \(x\) chess players. Use this formula to solve. In a round-robin chess tournament, 36 games were played. How many players were entered in the tournament?

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