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Chapter 3: Problem 33

Solve by completing the square to obtain exact solutions. $$x^{2}+8 x+25=0$$

Short Answer

Expert verified
The exact solutions are \( x = -4 + 3i \) and \( x = -4 - 3i \).

Step by step solution

01

- Move the constant term

Subtract 25 from both sides to move the constant term to the right side of the equation: \[ x^2 + 8x = -25 \]
02

- Complete the square

To complete the square, take half of the coefficient of x, square it, and add it to both sides of the equation. The coefficient of x is 8, half of 8 is 4, and 4 squared is 16: \[ x^2 + 8x + 16 = -25 + 16 \] So it becomes: \[ (x + 4)^2 = -9 \]
03

- Solve for x

Take the square root of both sides: \[ x + 4 = \pm \sqrt{-9} = \pm 3i \] So, the solutions are: \[ x = -4 \pm 3i \]

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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 polynomial equations of the form \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). These equations can have real or complex solutions.
The methods to solve them include factoring, using the quadratic formula, or completing the square.
  • Factoring: This involves expressing the quadratic equation as a product of its factors.
  • Quadratic Formula: The solutions can be found using \( x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \).
  • Completing the Square: This method involves rewriting the equation so that one side is a perfect square trinomial.
The exercise involves solving the equation by completing the square, which completes the square trinomial for easier extraction of the solution.
Complex Numbers
Complex numbers are numbers that have a real part and an imaginary part. They are written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\).
When we solve equations like \(x = -4 \, \pm \, 3i\), the term involving \(i\) represents a complex number.
  • The real part is the \(a\), representing the value without \(i\).
  • The imaginary part is the \(bi\), involving the imaginary unit \(i\).
  • The solutions of your quadratic equation involved complex numbers because of the negative inside the square root.
For this specific exercise, when you took the square root of \(-9\), it resulted in \( \pm 3i \). Hence, the solutions were complex numbers.
Mathematical Steps
The solution to this problem followed several careful mathematical steps. Understanding these will help you solve similar problems.
Step 1: Move the constant term:
You started with \(x^2 + 8x + 25 = 0\). Subtract 25 from both sides to isolate the quadratic term: \(x^2 + 8x = -25\).
Step 2: Complete the Square:
To complete the square, you took half of 8 (which is 4), squared it (getting 16), and added it to both sides: \(x^2 + 8x + 16 = -25 + 16\), forming \((x + 4)^2 = -9\).
Step 3: Solve for \(x\):
You took the square root of both sides: \( x + 4 = \pm 3i \), then subtracted 4 from both sides to get the final solution: \( x = -4 \, \pm \, 3i\).
  • Each mathematical step builds on the previous one to simplify and solve the equation.
  • Completing the square transforms the equation into a more solvable form.
  • Remembering each step will help demystify complex problems.

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