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Chapter 8: Problem 52

Solve by completing the square. Show your work. $$ x^{2}+8 x=9 $$

Short Answer

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Step by step solution

01

Move the constant to the right side

First, rewrite the equation by moving the constant term to the right side. This gives:

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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 an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. These equations often appear in algebra and have many applications in physics, engineering, and other sciences. In standard form:
  • \(a\) is the coefficient of \(x^2\) (it must not be zero)
  • \(b\) is the coefficient of \(x\)
  • \(c\) is the constant term
In the given exercise, the equation \(x^2 + 8x = 9\) is simplified from standard form by moving the constant term to the other side.
constant term
The constant term in a quadratic equation is the term that does not contain the variable \(x\). It is represented by \(c\) in the standard form \(ax^2 + bx + c = 0\). In the problem \(x^2 + 8x = 9\), the constant term is initially \(-9\) when we rearrange it into \(x^2 + 8x - 9 = 0\).
Moving the constant term to the right helps in simplifying the equation by isolating the variable. This is the first step you take when completing the square.
algebraic manipulation
Algebraic manipulation refers to the process of rearranging and simplifying algebraic expressions using basic operations like addition, subtraction, multiplication, and division. To solve \(x^2 + 8x = 9\) by completing the square, you need to use algebraic techniques to make the equation easier to solve.
Here鈥檚 a step-by-step approach to completing the square:
  • Step 1: Move the constant term to the right side: \(x^2 + 8x - 9 = 0\)
  • Step 2: Add a specific value to both sides to form a perfect square trinomial on the left
  • Step 3: Factor the perfect square trinomial
  • Step 4: Solve for \(x\) by taking the square root of both sides
Algebraic manipulation simplifies the process and helps to isolate the variable \(x\).
solving equations
Solving an equation means finding the value(s) of the variable that make the equation true. In the equation \(x^2 + 8x = 9\), solving involves completing the square. Let's continue from the previous steps:
After moving the constant term, we have \(x^2 + 8x = 9\). Add \(16\) (which is \((8/2)^2\)) to both sides to complete the square:
\(x^2 + 8x + 16 = 9 + 16\)
Now, the left side forms a perfect square trinomial:
\((x + 4)^2 = 25\)
Take the square root of both sides:
\(x + 4 = \pm 5\)
Finally, solve for \(x\):
\(x + 4 = 5\) gives \(x = 1\)
\(x + 4 = -5\) gives \(x = -9\)
Thus, the solutions are \(x = 1\) and \(x = -9\).

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