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Chapter 7: Problem 22

Find all real solutions to each equation. $$(w+3)^{2}-12=0$$

Short Answer

Expert verified
The solutions are \(w = -3 + 2\sqrt{3}\) and \(w = -3 - 2\sqrt{3}\).

Step by step solution

01

Isolate the Squared Term

Start by adding 12 to both sides of the equation to isolate the squared term. \[(w+3)^{2} - 12 + 12 = 0 + 12\]This simplifies to \[(w+3)^{2} = 12\]
02

Take the Square Root of Both Sides

Next, take the square root of both sides to solve for the expression inside the parentheses. \[\sqrt{(w+3)^{2}} = \pm \sqrt{12}\]This simplifies to \[w+3 = \pm \sqrt{12}\]Note that \(\sqrt{12}\) can be further simplified to \(2\sqrt{3}\). So, \[w + 3 = \pm 2\sqrt{3}\]
03

Solve for w

Isolate \(w\) by subtracting 3 from both sides. This gives two equations: \[w = -3 + 2\sqrt{3}\]\[w = -3 - 2\sqrt{3}\]

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Key Concepts

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

Isolating Terms
Isolating terms is a crucial first step in solving quadratic equations. When you isolate a term, you essentially move everything else to one side of the equation, leaving the term of interest by itself. This step makes it easier to manipulate the equation in subsequent steps.
For instance, in our problem $$(w+3)^{2} - 12=0$$, the goal is to isolate the squared term \((w+3)^{2} \). Here's how to do it:
  • Recognize the term to isolate: \( (w+3)^2 \).
  • Move the other terms to the opposite side of the equation. In this case, add 12 to both sides:
    \[ (w+3)^2 - 12 + 12 = 0 + 12 \] This simplifies to \[ (w+3)^2 = 12 \]
By isolating the squared term, the next steps of solving for \(w\) become much easier.
Square Root Method
The square root method involves taking the square root of both sides of an equation to solve for the variable. This step is essential after isolating a squared term. When applying the square root method:
  • Take the square root of both sides of the equation:
    \[ \sqrt{(w+3)^{2}} = \pm \sqrt{12} \]
  • Remember that taking the square root of a square term cancels out the square, leading to:
    \[ w+3 = \pm \sqrt{12} \]
It's crucial to consider both the positive and negative roots because squaring either a positive or negative number results in a positive value. In this case, \sqrt{12}\ can further be simplified, but for this step, focus on remembering to include the \( \pm \) symbol, representing both possible values.
Simplifying Square Roots
Simplifying square roots makes the final solution clearer and easier to interpret. Here's a step-by-step process to simplify a square root:
  • Identify factors of the number inside the square root that are perfect squares. For \sqrt{12}\, note that \12 = 4 \times 3\.
  • Rewrite the square root using these factors: \sqrt{12} = \sqrt{4 \times 3}\.
  • Simplify further by taking the square root of the perfect square: \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3} = 2 \sqrt{3}\.
Now rewrite the solutions using this simplified form:
\[ w+3 = \pm 2 \sqrt{3} \]
Finally, isolate \( w \) in both cases by subtracting 3 from both sides:
\[ w = -3 + 2 \sqrt{3} \] and \[ w = -3 - 2 \sqrt{3} \]

That's it! You've solved the quadratic equation using the square root method and simplified the square roots. Congratulations on mastering these core concepts!

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