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Chapter 11: Problem 58

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

Short Answer

Expert verified
x=1 or x=-9

Step by step solution

01

- Move the constant term

To complete the square, we first need to move the constant term to the right side of the equation: \[ x^{2} + 8x = 9. \]
02

- Find the term to complete the square

Take half of the coefficient of the linear term (8), square it, and add it to both sides of the equation. Half of 8 is 4, and 4 squared is 16: \[ x^{2} + 8x + 16 = 9 + 16. \]
03

- Rewrite the left side as a square

Rewrite the left side of the equation as a perfect square trinomial: \[ (x + 4)^{2} = 25. \]
04

- Solve for x

Take the square root of both sides and solve for x: \[ x + 4 = \pm 5. \] Therefore, \[ x = -4 + 5 \, \] \[ x = 1, \] or \[ x = -4 - 5 \, \] \[ x = -9. \]

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Key Concepts

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

Solving Quadratic Equations
Quadratic equations are equations of the form \( ax^2 + bx + c = 0 \). Solving them involves finding the values of \( x \) that make the equation true. One powerful method for solving quadratic equations is 'completing the square.' This technique transforms any quadratic equation into a perfect square trinomial, which can then be solved easily using the square root property.
Here's a step-by-step outline of how to solve a quadratic equation by completing the square:
  • Move the constant term to the opposite side of the equation.
  • Divide all terms by the coefficient of \( x^2 \) if it鈥檚 not 1.
  • Add and subtract the square of half the coefficient of x inside the equation. This creates a perfect square trinomial.
  • Rewrite the equation as the square of a binomial.
  • Take the square root of both sides of the equation.
    Don't forget to consider both the positive and negative square roots.
  • Finally, solve for \( x \).
Simplifying each step helps find the solutions quickly and efficiently.
Perfect Square Trinomial
A perfect square trinomial is an algebraic expression of the form \( (x + a)^2 \) or \( (x - a)^2 \). This means the expression can be written as \( x^2 + 2ax + a^2 \), where \( a \) is any constant number.
When completing the square, you manipulate your quadratic equation to form a perfect square trinomial. Here's how:
  • Identify the coefficient of the linear term (x-term).
  • Take half of this coefficient and square it.
  • Add and subtract this square inside your equation.
For example, in the equation \( x^2 + 8x = 9 \), the coefficient of \( x \) is 8. Half of 8 is 4, and 4 squared is 16. By adding and subtracting 16, you can rewrite the expression as \( x^2 + 8x + 16 - 16 = 9 \). This simplifies to \( (x + 4)^2 - 16 = 9 \), enabling you to solve the equation more easily.
Square Root Property
The square root property is a helpful tool for solving quadratic equations, especially after forming a perfect square trinomial. It states that if \( x^2 = k \), then \( x = \pm \sqrt{k} \).
Here's how to apply it:
  • First, isolate the square on one side of the equation.
  • Take the square root of both sides. Remember to consider both positive and negative roots.
  • Solve for the variable.
For example, with the equation \( (x + 4)^2 = 25 \), applying the square root property gives us:
\( x + 4 = \pm 5 \)
This splits into two separate equations:
\( x + 4 = 5 \)
\( x + 4 = -5 \)
Solving these, we find:
\( x = 1 \)
\( x = -9 \)
Thus, the solutions to the original quadratic equation are \( x = 1 \) and \( x = -9 \). By understanding the square root property, you can easily solve quadratic equations once they are in a perfect square trinomial form.

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Most popular questions from this chapter

Complete the square to find the \(x\) -intercepts of each function given by the equation listed. $$ g(x)=x^{2}+5 x+2 $$

Solve by completing the square. Show your work. $$ t^{2}-4 t=-1 $$

To prepare for Section \(11.2,\) review evaluating expressions and simplifying radical expressions (Sections \(1.8,10.3\) and \(10.8)\) Evaluate. [ 1.8] $$ b^{2}-4 a c, \text { for } a=3, b=2, \text { and } c=-5 $$

Complete each of the following to form a true statement. The principle of square roots states that if \(x^{2}=k\) then \(x=-\)_____ or \(x=\)_____

Write an equation for a function having a graph with the same shape as the graph of \(f(x)=\frac{3}{5} x^{2},\) but with the given point as the vertex. $$ (3,-1) $$
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