Problem 21 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Isolate the Square Term

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

Step 2: 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 \(x + 2 = -\sqrt{25}\).

Step 3: Evaluate the Square Root

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

Step 4: Solve for x

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

Key Concepts

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.