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Chapter 7: Problem 20

Many equations cannot be solved directly by backtracking. Some have the variable stated more than once; others involve variable as exponents. Here are some equations that can't be solved directly by backtracking. $$ \begin{array}{c}{f^{2}=f+1} & {x=\sqrt{x}+1 \quad k^{2}+k=0} \\ {1.1^{B}=2} & {\frac{1}{x}=x^{2}+2}\end{array} $$ For each equation below, write yes if it can be solved directly by backtracking and \(n o\) if it cannot. $$ \begin{array}{ll}{\text { a. } 5=\sqrt{x-11}} & {\text { b. } 4^{d}=9} \\\ {\text { c. } 3 g^{2}=5} & {\text { d. } \sqrt{x+1}=x-4}\end{array} $$

Short Answer

Expert verified
a. yes; b. no; c. yes; d. no

Step by step solution

01

- Analyze equation a

Consider the equation given: \[ 5 = \sqrt{x-11} \]Square both sides to eliminate the square root:\[ 5^2 = (\sqrt{x-11})^2 \]Simplify: \[ 25 = x-11 \]Solve for \( x \): \[ x = 36 \]This equation can be solved directly by squaring both sides. Therefore, the answer is 'yes'.
02

- Analyze equation b

Consider the equation: \[ 4^d = 9 \]We can take the logarithm of both sides to solve for \( d \): \[ \log(4^d) = \log(9) \]Using the power rule of logarithms: \[ d \log(4) = \log(9) \]Solve for \( d \): \[ d = \frac{\log(9)}{\log(4)} \]This requires the use of logarithms and cannot be solved by simple backtracking. Therefore, the answer is 'no'.
03

- Analyze equation c

Consider the equation: \[ 3g^2 = 5 \]Isolate \( g^2 \): \[ g^2 = \frac{5}{3} \]Take the square root of both sides:\[ g = \pm \sqrt{\frac{5}{3}} \]This equation can be solved directly by isolating and taking the square root. Therefore, the answer is 'yes'.
04

- Analyze equation d

Consider the equation: \[ \sqrt{x+1} = x-4 \]Square both sides to eliminate the square root:\[ (\sqrt{x+1})^2 = (x-4)^2 \]Simplify: \[ x+1 = x^2 - 8x + 16 \]Move all terms to one side of the equation:\[ x^2 - 9x + 15 = 0 \]Solve the quadratic equation, which involves factoring or using the quadratic formula. Therefore, the answer is 'no'.

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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 \(x\) is the variable to be solved. These equations are called 'quadratic' because 'quad' refers to the square, as the highest power of the variable \(x\) is 2.
There are a few methods to solve quadratic equations:
  • Factoring: This involves writing the quadratic equation as a product of two binomials. For example, \(x^2 - 9x + 15 = (x-3)(x-5)\).
  • Quadratic Formula: This formula can solve any quadratic equation: \[x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a}\]
  • Completing the Square: This method involves rearranging the equation to form a perfect square trinomial.
The quadratic formula is particularly useful when the equation cannot be easily factored.
When analyzing equation d from the exercise, we had \(x^2 - 9x + 15 = 0\). This quadratic equation cannot be solved by simple backtracking and requires factoring or the quadratic formula.
logarithms
Logarithms are the inverse operations of exponentiation. They tell us the power to which a number (the base) must be raised to obtain another number. The logarithm of a number \(y\) with base \(b\) is written as \(\log_b(y)\), and is defined as the exponent \(x\) such that \(b^x = y\).
The properties of logarithms include:
  • Product Rule: \(\log_b(xy) = \log_b(x) + \log_b(y)\)
  • Quotient Rule: \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\)
  • Power Rule: \(\log_b(x^k) = k \log_b(x)\)
In the exercise, the equation \(4^d = 9\) was analyzed by taking the logarithm of both sides and using the power rule, resulting in \(d\log(4) = \log(9)\). Therefore, isolating \(d\) required dividing by \(\log(4)\), which cannot be solved by simple backtracking.
square root
The square root of a number \(x\) is a value \(y\) such that \(y^2 = x\). In other words, if you square the square root, you get the original number. The square root function is denoted as \(\sqrt{x}\).
Here are a few important points about square roots:
  • The square root of a number is always non-negative, though some equations will consider both positive and negative roots.
  • When solving equations involving square roots, squaring both sides of the equation can help to eliminate the square root.
  • Remember to check for extraneous solutions when you square both sides, as squaring can introduce solutions that do not satisfy the original equation.
In the exercise, the equation \(5 = \sqrt{x-11}\) was solved by squaring both sides, resulting in \(25 = x-11\), and then isolating \(x\) to find \(x = 36\). This process is typical for equations involving square roots.

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Most popular questions from this chapter

Solve each equation by completing the square. $$ 2 x^{2}+4 x-1=0 $$

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