Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 5: Problem 15
Find all complex solutions for each equation by hand. $$\frac{4}{x^{2}-3 x}-\frac{1}{x^{2}-9}=0$$
Short Answer
Expert verified
The only solution is \( x = -4 \).
Step by step solution
01
Identify Common Denominator
Firstly, notice the expression we have, \(\frac{4}{x^{2}-3x} - \frac{1}{x^{2}-9} = 0\). The common denominator for these two fractions is the product of their denominators: \( (x^{2} - 3x)(x^{2} - 9) \).
02
Combine Fractions
Now, rewrite the equation with the common denominator: \[ \frac{4(x^{2}-9) - 1(x^{2}-3x)}{(x^{2}-3x)(x^{2}-9)} = 0 \].
03
Set the Numerator Equal to Zero
Since the denominator does not equal zero (so the fractions are defined), we can set the numerator equal to zero: \( 4(x^{2} - 9) - (x^{2} - 3x) = 0 \). Expand and combine the terms: \( 4x^{2} - 36 - x^{2} + 3x = 0 \). Simplify to get \( 3x^{2} + 3x - 36 = 0 \).
04
Simplify the Equation
Divide every term in the equation by 3 for simplicity: \( x^{2} + x - 12 = 0 \).
05
Factor the Quadratic Equation
Next, factor the quadratic \( x^{2} + x - 12 = 0 \). The solutions are \((x + 4)(x - 3) = 0\).
06
Solve for x
Set each factor equal to zero: \( x + 4 = 0 \) or \( x - 3 = 0 \). Thus, \( x = -4 \) and \( x = 3 \).
07
Check Solutions in the Original Denominator
We must check these solutions in the original denominators \( x^{2} - 3x \) and \( x^{2} - 9 \) to ensure they don鈥檛 make the original fractions undefined. For \( x = -4 \), \(x^{2} - 3x = 12\) and \(x^{2} - 9 = 7\). For \( x = 3 \), \(x^{2} - 3x = 0\) and \(x^{2} - 9 = 0\). Since \( x^{2} - 3x = 0\) for \(x = 3\), we discard this solution as it causes division by zero.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
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 second degree. They generally take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Solving quadratic equations involves finding the values of \(x\) that make the equation true. To solve these equations, several methods can be used:
By simplifying, factoring, or using the quadratic formula, solutions can be real or complex, depending on the discriminant \(b^2 - 4ac\). Here, we used factoring after simplifying the original equation to find the possible values for \(x\).
- Factoring: If the quadratic can be expressed as a product of two binomials, such as \((x + m)(x + n) = 0\), the solution is found by setting each factor equal to zero.
- Completing the square: This involves rewriting the quadratic equation in the form \((x + p)^2 = q\) to easily find the solutions.
- Quadratic formula: The formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) allows you to solve any quadratic equation by calculating the values of \(x\).
By simplifying, factoring, or using the quadratic formula, solutions can be real or complex, depending on the discriminant \(b^2 - 4ac\). Here, we used factoring after simplifying the original equation to find the possible values for \(x\).
Complex Numbers
Complex numbers extend our understanding of numbers by including a special unit \(i\), which is defined as \(i^2 = -1\). This allows us to work with equations involving the square root of negative numbers. A complex number is generally written as \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is the real part, and \(bi\) is the imaginary part. Some important properties of complex numbers include:
Complex solutions in quadratic equations arise when the discriminant \(b^2 - 4ac\) is negative, indicating no real solutions exist, only complex ones.
- Addition and Subtraction: Done by combining like terms. For example, \((3 + 4i) + (2 - 3i) = 5 + i\).
- Multiplication: Apply the distributive property and remember that \(i^2 = -1\). For instance, \((2 + 3i)(1 + 4i) = 2 + 8i + 3i + 12i^2 = -10 + 11i\).
- Division: Multiply the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary part from the denominator.
Complex solutions in quadratic equations arise when the discriminant \(b^2 - 4ac\) is negative, indicating no real solutions exist, only complex ones.
Algebraic Fractions
Algebraic fractions are fractions where the numerator, the denominator, or both involve algebraic expressions. Working with algebraic fractions includes finding common denominators to combine terms just as in numerical fractions.
Key steps for solving equations with algebraic fractions:
In quadratic equations, as with our exercise example, care must be taken to avoid solutions that make any part of the original denominator zero, ensuring valid solutions.
- Identify Common Denominator: Combine fractions by rewriting them with a common denominator. This was done in the exercise to simplify and solve the equation.
- Simplify: After combining the fractions, simplify the equation by canceling common factors or reducing expressions if possible.
- Solving: Often involves setting the simplified numerator equal to zero after ensuring the denominator does not equal zero, avoiding undefined expressions.
In quadratic equations, as with our exercise example, care must be taken to avoid solutions that make any part of the original denominator zero, ensuring valid solutions.
