Problem 49 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

x = 8 or x = -2.

Step 1: Understand the Square Root Property

The square root property states that if \[ a^2 = b \] then \[ a = \pm \sqrt{b} \]. This will be applied to solve the quadratic equation.

Step 2: Take the Square Root of Both Sides

Given equation: \[ (x-3)^2 = 25 \]. Taking the square root of both sides gives: \[ x-3 = \pm \sqrt{25} \]. Simplifying the right-hand side yields: \[ x-3 = \pm 5 \].

Step 3: Solve for x

Now, set up two separate equations to solve for x: \[ x-3 = 5 \] and \[ x-3 = -5 \]. Solve each equation: 1. \[ x-3 = 5 \] \[ x = 5 + 3 \] \[ x = 8 \]. 2. \[ x-3 = -5 \] \[ x = -5 + 3 \] \[ x = -2 \].

Step 4: Verify the Solutions

Check the solutions by substituting x = 8 and x = -2 back into the original equation: 1. For x = 8: \[ (8-3)^2 = 25 \] \[ 5^2 = 25 \] \[ 25 = 25 \] 2. For x = -2: \[ (-2-3)^2 = 25 \] \[ (-5)^2 = 25 \] \[ 25 = 25 \]. Since both solutions satisfy the original equation, they are correct.

Key Concepts

square root property

When solving quadratic equations, we often use a handy technique known as the Square Root Property. This property makes it easier to find the unknown value by transforming the equation into a simpler form. The Square Root Property says that for any number 'a', if \[ a^2 = b \], then \[ a = \pm \sqrt{b} \]. This means that 'a' can be either the positive or negative square root of 'b'.
For example, if you have \[ (x-3)^2 = 25 \], you can apply the square root to both sides of the equation. Doing this, you get \[ x-3 = \pm \sqrt{25} \]. Since the square root of 25 is 5, the equation becomes \[ x-3 = \pm 5 \]. This step simplifies and reduces the equation to two possible simple forms: \[ x-3=5 \] and \[ x-3=-5 \].
Understanding this property is the key when dealing with quadratic equations involving squared terms as it allows us to break down a complex equation into simpler, more manageable parts.

simplifying radicals

Simplifying radicals is a crucial step when solving equations involving square roots. A radical expression can often be simplified to its simplest form, which makes the equation easier to solve. When you simplify a radical, you aim to find the most basic square root of a number.
For instance, when solving the equation \[ (x-3)^2 = 25 \], you take the square root of both sides to get \[ x-3 = \pm \sqrt{25} \]. The square root of 25 is 5 because \[ 5^2 = 25 \]. Therefore, the expression \sqrt{25} \ can be simplified to 5.
Here are a few steps to simplify radicals:

• Find the prime factors of the number under the radical.
• Pair the prime factors in groups of two (because \[ \sqrt{a \cdot a} = a \]).
• Take one number from each pair out of the radical.

By using these steps, you ensure that the radical is in the simplest form, making it easier to solve the remaining equation.

verifying solutions

Once you've solved the equation, it's crucial to verify each solution by substituting it back into the original equation to ensure it holds true. This step confirms the accuracy of your solutions.
To verify the solutions for \[ (x-3)^2 = 25 \], we found two possibilities: \[ x = 8 \] and \[ x = -2 \].

• For \[ x = 8 \]: Substitute '8' back into the original equation. \[ (8-3)^2 = 25 \]. This simplifies to \[ 5^2 = 25 \], which is true since \[ 25 = 25 \].
• For \[ x = -2 \]: Substitute '-2' back into the original equation. \[ (-2-3)^2 = 25 \]. This simplifies to \[ (-5)^2 = 25 \], which is also true since \[ 25 = 25 \].

Since both solutions satisfy the original equation, they are considered correct. Verifying solutions is a crucial step in solving equations as it ensures that the values you find genuinely solve the problem.