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Magazine Chapter 7: Problem 35
For Problems \(33-50\), set up an equation and solve the problem. (Objective 2 ) $$ \text { A number is } \frac{21}{10} \text { larger than its reciprocal. Find the number. } $$
Short Answer
Expert verified
The number is 2.5.
Step by step solution
01
Define the Variables
Let the number be represented by \( x \). Its reciprocal will then be \( \frac{1}{x} \).
02
Set up the Equation
According to the problem, the number \( x \) is \( \frac{21}{10} \) larger than its reciprocal. This can be expressed as:\[ x = \frac{1}{x} + \frac{21}{10} \]
03
Eliminate the Fraction
To eliminate the fraction, multiply every term by \( x \) (assuming \( x eq 0 \)) to obtain:\[ x^2 = 1 + \frac{21}{10}x \]
04
Clear the Equation
Multiply each term by 10 to remove the fraction:\[ 10x^2 = 10 + 21x \]
05
Rearrange into a Quadratic Equation
Rearrange the equation to bring all terms to one side:\[ 10x^2 - 21x - 10 = 0 \]
06
Solve the Quadratic Equation
Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 10 \), \( b = -21 \), and \( c = -10 \):- Calculate the discriminant: \( b^2 - 4ac = 441 + 400 = 841 \).- Find the roots: \( x = \frac{21 \pm \sqrt{841}}{20} \).- Simplify to get \( x = \frac{21 \pm 29}{20} \): - \( x = \frac{50}{20} = 2.5 \) - \( x = \frac{-8}{20} = -0.4 \).
07
Verify the Solutions
Check if both solutions satisfy the condition in the original problem:- For \( x = 2.5 \), \( \frac{1}{x} = \frac{1}{2.5} = 0.4 \), and \( 2.5 = 0.4 + \frac{21}{10} \) which is true.- For \( x = -0.4 \), \( \frac{1}{x} = -2.5 \), and \( -0.4 eq -2.5 + \frac{21}{10} \). So, this does not hold.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Reciprocals
In mathematics, a reciprocal is essentially what you get when you exchange the numerator and the denominator of a fraction. For any non-zero number \( x \), the reciprocal is \( \frac{1}{x} \). This simple operation is quite significant because when you multiply a number by its reciprocal, the result is always 1. That's the defining property:
- For example, the reciprocal of 5 is \( \frac{1}{5} \), and \( 5 \times \frac{1}{5} = 1 \).
- Similarly, the reciprocal of \( \frac{2}{3} \) is \( \frac{3}{2} \), and \( \frac{2}{3} \times \frac{3}{2} = 1 \).
The Role of the Discriminant
The discriminant is an integral part of the quadratic formula and plays a crucial role in determining the nature of the roots of a quadratic equation. For any quadratic equation in the form \( ax^2 + bx + c = 0 \), the discriminant is given by the expression \( b^2 - 4ac \). The value of this discriminant indicates:
- When \( b^2 - 4ac > 0 \), the quadratic equation has two distinct real roots.
- If \( b^2 - 4ac = 0 \), the equation has exactly one real root, known as a repeated or double root.
- And if \( b^2 - 4ac < 0 \), the equation has no real roots but two complex conjugate roots.
Utilizing the Quadratic Formula
The quadratic formula is a crucial tool for solving quadratic equations and can be used for any quadratic equation in the form of \( ax^2 + bx + c = 0 \). It is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula provides the solutions for \( x \) by using the coefficients \( a \), \( b \), and \( c \). Here's how to apply it effectively:
- Plug the values for \( a \), \( b \), and \( c \) into the formula.
- Calculate the discriminant \( b^2 - 4ac \) to discover the number and type of solutions.
- Apply the solutions formula to solve for \( x \).
