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Chapter 9: Problem 67

Solve using the Square Root Property. $$(x-6)^{2}+7=3$$

Short Answer

Expert verified
The solutions are \(x = 6 + 2i\) and \(x = 6 - 2i\).

Step by step solution

01

- Isolate the squared term

First, subtract 7 from both sides of the equation to isolate the squared term. \[(x-6)^2 + 7 - 7 = 3 - 7\] Simplify to get: \[(x-6)^2 = -4\]
02

- Recognize no real solution

Notice that \((x-6)^2\) represents a square of a real number, which cannot equal a negative number like \(-4\). In real numbers, there is no solution. However, in the complex number system, proceed by taking the square root of both sides.
03

- Take the square root of both sides

Apply the square root property. When taking the square root of both sides, consider both the positive and negative square roots: \[\sqrt{(x-6)^2} = \pm \sqrt{-4}\] This simplifies to: \[x-6 = \pm 2i\]
04

- Solve for x

Add 6 to both sides to isolate \(x\): \[x-6 + 6 = \pm 2i + 6\] So, \[x = 6 \pm 2i\]

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Key Concepts

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

complex numbers
Complex numbers come into play when we deal with the square roots of negative numbers.
The equation \( (x-6)^2 = -4 \) brings us into the realm of complex numbers because the square of a real number cannot be negative.

To solve such equations, we use the imaginary unit \(i\), where \(i^2 = -1\).
Thus, \( \sqrt{-4} = 2i \).
Complex numbers are represented as \ a + bi \, where \(a\) and \(b\) are real numbers.
solving quadratic equations
Quadratic equations can be solved using various methods like factoring, the quadratic formula, completing the square, and the square root property.
In this case, \( (x-6)^{2}+7 = 3 \, we use the square root property.

The steps typically include:
  • Isolating the squared term
  • Taking the square root of both sides, considering both the positive and negative roots
  • Solving for the variable
Applying these steps to our equation, we isolate the squared term to get \ (x-6)^{2} = -4 \).
Then, we take the square root to find \ x = 6 \pm 2i \.
isolation of terms
Isolating terms is a crucial step in solving equations, especially when using the square root property.
Let's look at the first step of the solution: subtracting 7 from both sides of \( (x-6)^2 + 7 = 3 \).

This simplifies to \ (x-6)^2 = -4 \, effectively isolating the squared term.
By isolating the term \( (x-6)^2 \), we can clearly see that we need to find values of \ x \ by taking the square root of both sides.
Always remember to perform the same operation on both sides of the equation to maintain balance.
Once isolated, we can then proceed with solving the resulting equation.

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Most popular questions from this chapter

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Graph each function using transformations. $$f(x)=(x+2)^{2}+1$$

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Graph each function using a horizontal shift. $$f(x)=(x-2)^{2}$$
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