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Chapter 11: Problem 21

Solve the given equation by the method of completing the square. $$x^{2}+4 x+5=2 x-3$$

Short Answer

Expert verified
The solutions are \( x = -1 + \sqrt{7}i \) and \( x = -1 - \sqrt{7}i \).

Step by step solution

01

- Move All Terms to One Side

First, move all terms to one side of the equation to set it equal to zero: \[ x^2 + 4x + 5 - 2x + 3 = 0 \]Simplify the equation: \[ x^2 + 2x + 8 = 0 \]
02

- Isolate Constant Term

Next, isolate the constant term by moving it to the other side of the equation: \[ x^2 + 2x = -8 \]
03

- Find the Term to Complete the Square

To complete the square, take half of the coefficient of the x-term (which is 2), square it, and add it to both sides of the equation. Half of 2 is 1, and 1 squared is 1: \[ x^2 + 2x + 1 = -8 + 1 \]This simplifies to: \[ x^2 + 2x + 1 = -7 \]
04

- Write the Left Side as a Perfect Square

The left-hand side can now be written as a perfect square: \[ (x + 1)^2 = -7 \]
05

- Solve for x

Take the square root of both sides to solve for x, remembering to include the 卤 symbol: \[ x + 1 = \pm \sqrt{-7} \]Since \( \sqrt{-7}\) is an imaginary number, write it as \(\sqrt{7}i\): \[ x + 1 = \pm \sqrt{7}i \]Lastly, solve for x: \[ x = -1 \pm \sqrt{7}i \]

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Key Concepts

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

Quadratic Equations
Quadratic equations are fundamental in algebra. They take the general form of \(ax^2 + bx + c = 0\), where \(a, b, \) and \(c\) are constants. The solutions to these equations can be found using various methods, such as factoring, using the quadratic formula, or completing the square. In our problem, we used the completing the square method to solve the quadratic equation. By rearranging and manipulating the equation's terms, we transform it into a form that allows us to find the solutions more easily. This process reveals the roots of the quadratic and offers parameters, including real and imaginary parts.
Imaginary Numbers
Imaginary numbers extend our number system into new territories. The unit imaginary number is \(i\), where \(i^2 = -1\). Whenever we encounter the square root of a negative number, it is written in terms of \(i\). In the solution of our quadratic equation, we ended up with the square root of -7. Hence, we represented it as \(\root{7}i\), indicating that the solutions involve imaginary numbers. Imaginary numbers are crucial in advanced mathematics, physics, and engineering because they allow calculations that would otherwise be impossible within the realm of real numbers.
Algebraic Methods
Algebraic methods provide tools to manipulate and solve equations. In our exercise, we used the method of completing the square. Here鈥檚 a concise summary of how this method works:
  • Move all terms to one side of the equation to set it equal to zero.
  • Isolate the constant term by moving it to the other side.
  • Find the term needed to complete the square by taking half the coefficient of the \(x\)-term, squaring it, and adding it to both sides.
  • Rewrite the left-hand side as a perfect square trinomial.
  • Solve by taking the square root of both sides and isolating \(x\).
These steps not only help solve the equation but also ensure a deep understanding of algebraic structure and methods in problem-solving.

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Most popular questions from this chapter

Multiply and simplify. $$(x-5)^{2}$$

Simplify as completely as possible. (Assume \(x \geq 0 .)\) $$\frac{12}{\sqrt{5}-\sqrt{3}}$$

In Exercises \(65-74\), solve the given equation. For quadratic equations, choose either the factoring method or the square root method, whichever you think is the easier to use. $$x^{2}+4 x+9=3 x^{2}+4 x+1$$

Suppose that a company's profit \(P,\) in thousands of dollars, on the sale of \(x\) items is given by the equation. $$P=x^{2}-6 x-40 \quad \text { where } x \geq 0$$ (a) Sketch the graph of this equation. Let \(x\) be the horizontal axis and \(P\) be the vertical axis. (b) Use the graph to determine the minimum profit the company earns. (c) How many items does the company sell if it experiences this minimum profit?

Sketch the graphs of \(y=x^{2}, y=2 x^{2},\) and \(y=3 x^{2}\) on the same coordinate system. How would you describe the effect the coefficients 2 and 3 have on the graph of \(y=x^{2} ?\)
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