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Chapter 1: Problem 16

Which of the following are solutions to the quadratic equation \((x-1)^{2}=\frac{4}{9} ?\) $$ \begin{array}{l}{\text { (A) } x=-\frac{5}{3}, x=\frac{5}{3}} \\ {\text { (B) } x=\frac{1}{3}, x=\frac{5}{3}} \\ {\text { (C) } x=\frac{5}{9}, x=\frac{13}{9}} \\ {\text { (D) } x=1 \pm \sqrt{\frac{2}{3}}}\end{array} $$

Short Answer

Expert verified
Choice (B): x = \frac{1}{3}, x = \frac{5}{3}

Step by step solution

01

Understand the given equation

The given quadratic equation is \( (x-1)^2 = \frac{4}{9} \). This means we need to find x values that satisfy this equation.
02

Take the square root on both sides

To eliminate the square on the left side, take the square root of both sides: \( \sqrt{(x-1)^2} = \sqrt{\frac{4}{9}} \). Simplifying this gives us two potential solutions: \( x - 1 = \frac{2}{3} \) and \( x - 1 = -\frac{2}{3} \).
03

Solve for x

Solve \( x - 1 = \frac{2}{3} \) by adding 1 to both sides: \( x = \frac{2}{3} + 1 = \frac{5}{3} \). Now, solve \( x - 1 = -\frac{2}{3} \) by adding 1 to both sides: \( x = -\frac{2}{3} + 1 = \frac{1}{3} \).
04

Compare solutions to given answer choices

Review the possible solutions and compare them to the given answer choices: \( x = \frac{5}{3} \) and \( x = \frac{1}{3} \) according to the answers are found in answer choice (B).

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Key Concepts

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

solving quadratic equations
Solving quadratic equations is a fundamental algebraic task.
A quadratic equation typically takes the form of \(ax^2 + bx + c = 0\).
To solve it, we can factorize, complete the square, or use the quadratic formula.
In this particular exercise, the equation is already simplified as \((x-1)^2 = \frac{4}{9}\).
This makes it easier to use the square root method to find the solutions.
Remember, solving quadratic equations involves finding values of x that satisfy the equation.
taking square roots
To solve \((x-1)^2 = \frac{4}{9}\) using square roots, follow these steps:
  • First, recognize that both sides of the equation need to be positive to take the square root.
  • Taking the square root on both sides, we have \sqrt{(x-1)^2} = \sqrt{\frac{4}{9}}\.
  • Since square roots have both positive and negative solutions, this results in two equations: \(x-1 = \frac{2}{3}\) and \(x-1 = -\frac{2}{3}\).

These steps convert the squared term into manageable linear equations, paving the way to find the values of x.
simplifying algebraic expressions
Simplifying algebraic expressions involves combining like terms and performing arithmetic operations.
In this problem, after taking the square roots, we get two simplified linear equations:
  • \(x - 1 = \frac{2}{3}\)
  • \(x - 1 = -\frac{2}{3}\)
To solve these, add 1 to both sides:
  • \(x = \frac{2}{3} + 1\)
  • \(x = -\frac{2}{3} + 1\)
This results in two simplified expressions for x:
  • \(x = \frac{5}{3}\)
  • \(x = \frac{1}{3}\)
These values are the solutions to the quadratic equation.
comparing solutions
Now, compare the solutions obtained from your equation with the provided answer choices.
  • The values we obtained are \(x = \frac{5}{3}\) and \(x = \frac{1}{3}\).
  • Check these values against the options provided in the problem statement.
We see that answer choice (B) matches our solutions:
\(x=\frac{1}{3}, x=\frac{5}{3}\). This confirms that our solutions and calculations are correct.
Comparing solutions helps in verifying the accuracy and ensuring the correct answers are chosen.

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Most popular questions from this chapter

$$\frac{1}{x}+\frac{4}{x}=\frac{1}{72}$$ In order to create safe drinking water, cities and towns use water treatment facilities to remove contaminants from surface water and groundwater. Suppose a town has a treatment plant but decides to build a second, more efficient facility. The new treatment plant can filter the water in the reservoir four times as quickly as the older facility. Working together, the two facilities can filter all the water in the reservoir in 72 hours. The equation above represents the scenario. Which of the following describes what the term \(\frac{1}{x}\) represents? (A) The portion of the water the older treatment plant can filter in 1 hour (B) The time it takes the older treatment plant to filter the water in the reservoir (C) The time it takes the older treatment plant to filter \(\frac{1}{72}\) of the water in the reservoir (D) The portion of the water the new treatment plant can filter in 4 hours

What value of \(x\) satisfies the equation $$\frac{2}{3}(5 x+7)=8 x?$$

In the United States, the maintenance and construction of airports, transit systems, and major roads is largely funded through a federal excise tax on gasoline. Based on the 2011 statistics given below, how much did the average household pay per year in federal gasoline taxes? $$\begin{array}{l}{\bullet \text { The federal gasoline tax rate was } 18.4 \text { cents per }} \\ {\text { gallon. }} \\ {\bullet \text { The average motor vehicle was driven }} \\ {\text { approximately } 11,340 \text { miles per year. }} \\ {\bullet \text { The national average fuel economy for }} \\\ {\text { noncommercial vehicles was } 21.4 \text { miles per gallon. }} \\\ {\bullet\text { The average American household owned }} \\ {\text { 1.75 vehicles. }}\end{array}$$ $$\begin{array}{l}{\text { (A) } \$ 55.73} \\ {\text { (B) } \$ 68.91} \\\ {\text { (C) } \$ 97.52} \\ {\text { (D) } \$ 170.63}\end{array}$$

$$\frac{3.86}{x}+\frac{180.2}{10 x}+\frac{42.2}{5 x}$$ The Ironman Triathlon originated in Hawaii in \(1978 .\) The format of the Ironman has not changed since then: It consists of a \(3.86-\mathrm{km}\) swim, a \(180.2-\mathrm{km}\) bicycle ride, and a \(42.2-\mathrm{km}\) run, all raced in that order and without a break. Suppose an athlete bikes 10 times as fast as he swims and runs 5 times as fast as he swims. The variable \(x\) in the expression above represents the rate at which the athlete swims, and the whole expression represents the number of hours that it takes him to complete the race. If it takes him 16.2 hours to complete the race, how many kilometers did he swim in 1 hour? $$\begin{array}{l}{\text { (A) } 0.85} \\ {\text { (B) } 1.01} \\ {\text { (C) } 1.17} \\ {\text { (D) } 1.87}\end{array}$$

$$\left\\{\begin{array}{l}{-2 x+5 y=1} \\ {7 x-10 y=-11}\end{array}\right.$$ If \((x, y)\) is a solution to the system of equations above, what is the sum of \(x\) and \(y ?\) $$ \begin{array}{l}{\text { (A) }-\frac{137}{30}} \\ {\text { (B) }-4} \\\ {\text { (C) }-\frac{10}{3}} \\ {\text { (D) }-3}\end{array} $$
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