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Chapter 10: Problem 79

Solve by completing the square. \(r^{2}+6 r=-11\)

Short Answer

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

Step by step solution

01

Move the constant term to the other side

First, rearrange the equation to isolate the constant term on one side. Start with the given equation: \( r^{2} + 6r = -11 \) Add 11 to both sides: \( r^{2} + 6r + 11 = 0 \) \( r^{2} + 6r = -11 \)
02

Complete the square

To complete the square, add and subtract the square of half the coefficient of r. The coefficient of r is 6. Half of 6 is 3, and squaring it gives 9. Add and subtract 9 to the left side: \( r^{2} + 6r + 9 - 9 = -11 \) \( r^{2} + 6r + 9 = -11 + 9 \) \( r^{2} + 6r + 9 = -2 \)
03

Simplify the left side into a binomial square

Notice that the left side is a perfect square trinomial. It simplifies to: \( (r + 3)^{2} = -2 \)
04

Solve for r

Take the square root of both sides to solve for r: \( r + 3 = \pm \sqrt{-2} \) Since the right side has a negative number under the square root, it gives imaginary solutions: \( r + 3 = \pm i\sqrt{2} \) Finally, isolate r by subtracting 3 from both sides: \( r = -3 \pm i\sqrt{2} \)

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Key Concepts

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

Quadratic Equations
A quadratic equation is any equation in the form of \( ax^2 + bx + c = 0 \). The solution to a quadratic equation can be real or imaginary. In our example, we start with \( r^2 + 6r = -11 \). The main goal here is to bring it to a workable form so we can solve for \( r \). This involves moving terms around and using algebraic techniques, like completing the square, to simplify.
Imaginary Numbers
Imaginary numbers come into play when you run into the square root of a negative number. For instance, \( \sqrt{-1} \) is defined as \( i \), the imaginary unit. In the provided solution, after completing the square, we end up with \( (r + 3)^2 = -2 \). Taking the square root of -2 introduces imaginary numbers, resulting in \( r + 3 = \pm i\sqrt{2} \). This means our solutions involve imaginary parts: \( r = -3 \pm i\sqrt{2} \).
Algebraic Techniques
When solving quadratic equations, a variety of algebraic techniques can be used. One of the most efficient methods is completing the square. This process works by converting the quadratic equation into a form that can easily be solved by taking the square root. Steps in our example included isolating the constant term and then converting the quadratic expression into a perfect square trinomial: \( r^2 + 6r + 9 \), which simplifies to \( (r + 3)^2 \).
Binomial Square
A binomial square is a useful algebraic expression of the form \( (r + k)^2 \) or \( (r - k)^2 \). It comes from the expansion \( r^2 + 2kr + k^2 \). In our case, after completing the square, we achieved \( (r+3)^2 \) = \( r^2 + 6r + 9 \). This method simplifies solving for r when the left side of the equation becomes a neat binomial square, making it ready for taking the square root.

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