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Chapter 8: Problem 71

Solve each equation by completing the square. \(z^{2}-\frac{4}{3} z=-\frac{1}{9}\)

Short Answer

Expert verified
\[ z = \frac{2 \pm \sqrt{5}}{3} \]

Step by step solution

01

- Move the constant term

First, move the constant term to the other side of the equation to prepare for completing the square. Starting equation: \[ z^{2} - \frac{4}{3}z = -\frac{1}{9} \] Move the constant term: \[ z^{2} - \frac{4}{3}z + \frac{1}{9} = 0 \] Rewriting gives: \[ z^{2} - \frac{4}{3}z = \frac{1}{9} \]
02

- Add and subtract the required term

Next, find the term that completes the square. The term needed is \[ \left(\frac{b}{2}\right)^2 \] where \( b = -\frac{4}{3} \). This term is \[ \left(\frac{-\frac{4}{3}}{2}\right)^2 = \left(-\frac{2}{3}\right)^2 = \frac{4}{9} \]. Add and subtract this term inside the equation: \[ z^{2} - \frac{4}{3}z + \frac{4}{9} - \frac{4}{9} = \frac{1}{9} \].
03

- Form a perfect square trinomial

Convert the left-hand side into a perfect square trinomial and simplify the right-hand side: \[ (z - \frac{2}{3})^{2} - \frac{4}{9} = \frac{1}{9} \]. Combine the constants on the right-hand side: \[ (z - \frac{2}{3})^{2} = \frac{1}{9} + \frac{4}{9} \]. \[ (z - \frac{2}{3})^{2} = \frac{5}{9} \].
04

- Solve for z

Take the square root of both sides to solve for \(z\): \[ z - \frac{2}{3} = \pm \sqrt{\frac{5}{9}} \]. \[ z - \frac{2}{3} = \pm \frac{\sqrt{5}}{3} \]. Add \( \frac{2}{3} \) to both sides: \[ z = \frac{2}{3} \pm \frac{\sqrt{5}}{3} \]. This simplifies to \[ z = \frac{2 \pm \sqrt{5}}{3} \].

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Key Concepts

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

quadratic equation
A quadratic equation is a type of algebraic equation of the form \( ax^2 + bx + c = 0 \). Here, \( a, b, \) and \( c \) are constants with \( a eq 0 \). Quadratic equations are essential in algebra and have various applications in real life, from physics to finance.
The standard formula to solve a quadratic equation is called the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
However, another method to solve these equations is by *completing the square*, which simplifies the process and helps in better understanding concepts like the vertex form of a parabola.
solving equations
Solving equations means finding the values of the variable that make the equation true. In the context of quadratic equations, we're looking for the values of \( z \) that satisfy the quadratic equation.
To solve a quadratic equation by completing the square, you follow these key steps:
  • Move the constant term to the other side of the equation.
  • Add and subtract a specific value to turn the left-hand side into a perfect square trinomial.
  • Factor the perfect square trinomial and simplify the other side of the equation.
  • Solve for the variable by taking the square root of both sides and isolating the variable.
perfect square trinomial
A perfect square trinomial is an expression of the form \( (x + y)^2 \) or \( (x - y)^2 \), which expands to \( x^2 + 2xy + y^2 \) and \( x^2 - 2xy + y^2 \), respectively.
In the problem, we transformed the equation into a perfect square trinomial: \( (z - \frac{2}{3})^2 \).
To create a perfect square trinomial, you need to find the term \( \left( \frac{b}{2} \right)^2 \). By completing the square, we transformed \( z^2 - \frac{4}{3}z \) into \( (z - \frac{2}{3})^2 \).
It's a powerful method that simplifies the expression and makes solving the equation straightforward.

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