Problem 45 - Elementary Algebra

Question

Solve using the Square Root Property. \((m-4)^{2}+3=15\)

Step-by-Step Solution

Answer

The solutions are \( m = 4 + 2 \sqrt{3} \) and \( m = 4 - 2 \sqrt{3} \).

Step 1: Isolate the squared term

Start by isolating the squared term \( (m-4)^2 \). To do this, subtract 3 from both sides of the equation: \( (m-4)^2 + 3 - 3 = 15 - 3 \) This simplifies to: \( (m-4)^2 = 12 \)

Step 2: Apply the Square Root Property

Next, apply the Square Root Property. Take the square root of both sides of the equation to solve for \( m - 4 \): \[ \sqrt{(m-4)^2} = \sqrt{12} \] Simplifying this further gives: \[ m - 4 = \pm \sqrt{12} \] Since \( \sqrt{12} \) can be simplified to \( 2 \sqrt{3} \): \[ m - 4 = \pm 2 \sqrt{3} \]

Step 3: Solve for m

Finally, solve for \( m \) by adding 4 to both sides of the equation: \[ m = 4 \pm 2 \sqrt{3} \] This gives us two solutions: \[ m = 4 + 2 \sqrt{3} \] and \[ m = 4 - 2 \sqrt{3} \]

Key Concepts

Square Root Property

The Square Root Property is a useful tool for solving equations where the variable is squared. It simply states that if you have an equation of the form ewline \[ x^2 = k \], ewline then you can take the square root of both sides, resulting in \[ x = \pm \sqrt{k} \]. Remember, you always get two solutions because both positive and negative numbers squared give the same result. In our exercise, after isolating the squared term, we used this property on ewline \[ (m-4)^2 = 12 \], ewline giving us ewline \[ m-4 = \pm \sqrt{12} \].

Isolating Terms

Isolating the term that includes the variable you want to solve for is a crucial first step. For our problem, we had the equation ewline \[ (m-4)^2 + 3 = 15 \]. ewline We needed to isolate ewline \[ (m-4)^2 \], ewline so we subtracted 3 from both sides: ewline \[ (m-4)^2 = 12 \]. ewline This simplifies the problem, making it easier to apply further steps like the Square Root Property. The goal is to have the term with the variable by itself on one side of the equation.

Simplifying Radicals

When you take square roots, you often get numbers that are not whole, known as radicals. Simplifying radicals is important to find the cleanest form of the solution. In our example, the square root of 12 simplifies further. ewline We know: ewline \[ 12 = 4 \times 3 \] ewline and ewline \[ \sqrt{4} = 2 \], ewline so ewline \[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2 \sqrt{3} \]. ewline Therefore, ewline \[ m - 4 = \pm 2 \sqrt{3} \]. ewline Simplifying radicals makes the final answer more understandable and elegant.

Step-by-Step Solutions

Breaking down problems into manageable steps is key. Here is a recap of our process for solving ewline \[ (m-4)^2 + 3 = 15 \]: ewline ewline ewline

• Step 1: Isolate the squared term by subtracting 3: ewline \[ (m-4)^2 = 12 \].
• Step 2: Apply the Square Root Property by taking the square root of both sides: ewline \[ m-4 = \pm \sqrt{12} \] ewline and simplify to ewline \[ m-4 = \pm 2 \sqrt{3} \].
• Step 3: Solve for ewline \[ m \] ewline by adding 4: ewline \[ m = 4 \pm 2 \sqrt{3} \].

ewline Always working step-by-step, we ensure we understand and solve parts of the problem before moving on.