Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 10: Problem 35
Solve the fractional equation \(\frac{4}{x+12}+\frac{3}{x+3}=\frac{4}{x^{2}+5 x+6} .\)
Short Answer
Expert verified
Answer: The two solutions to the given fractional equation are x = -2 and x = -6.
Step by step solution
01
Identify the common denominator
The common denominator or least common multiple (LCM) of the given denominators \((x+12)\), \((x+3)\) and \((x^2+5x+6)\) can be found by factoring the last polynomial, which can be factored as \((x + 3)(x + 2)\). Therefore, the LCM is \((x + 12)(x + 3)(x + 2)\).
02
Multiply each term by the LCM
Now we will multiply each term of the equation by the LCM, to eliminate the denominators:
\[(x + 12)(x + 3)(x + 2) \cdot \frac{4}{(x+12)} \,+\, (x + 12)(x + 3)(x + 2) \cdot \frac{3}{(x+3)} \,=\, (x + 12)(x + 3)(x + 2) \cdot \frac{4}{(x^2+5x+6)}\]
03
Simplify the equation
Next, we simplify the equation by canceling out the common factors:
\[(x + 3)(x + 2) \cdot 4 + (x + 12)(x + 2) \cdot 3 = 4(x + 12)\]
Now expand and simplify:
\[4(x^2 + 5x + 6) + 3(x^2 + 14x + 24) = 4x + 48\]
Combining the terms, we get:
\[7x^2 + 56x + 72 = 4x + 48\]
04
Solve for the variable
Now, we need to solve for \(x\). First, move all terms to one side of the equation:
\[7x^2 + 52x + 24 = 0\]
This is a quadratic equation. To solve it, we can either factorize it, use the quadratic formula, or find the roots by completing the square. In this case, we'll use the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
Plugging in the values:
\[x = \frac{-52 \pm \sqrt{52^2 - 4 \cdot 7 \cdot 24}}{2 \cdot 7}\]
Calculate the discriminant:
\[D = 52^2 - 4 \cdot 7 \cdot 24 = 2304\]
Now we have:
\[x = \frac{-52 \pm \sqrt{2304}}{14}\]
Find the two possible values of \(x\):
\[x_1 = \frac{-52 + \sqrt{2304}}{14} = -2\]
\[x_2 = \frac{-52 - \sqrt{2304}}{14} = -6\]
So the solutions to the given fractional equation are \(x = -2\) and \(x = -6\).
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 Equation
A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The presence of the \(x^2\) term makes the equation quadratic. Quadratic equations can have two solutions because they represent a parabola that can intersect the x-axis at two points. There are several methods to solve quadratic equations, such as:
- Factoring: This involves expressing the quadratic as a product of two binomials and finding the values of \(x\) that make each binomial zero.
- Quadratic Formula: This formula is used when factoring is difficult or impossible. It is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
- Completing the Square: This involves rearranging the equation and adding a term to both sides to make a perfect square trinomial.
Factoring Polynomials
Factoring polynomials is the process of expressing a polynomial as a product of its factors. It is particularly useful for simplifying algebraic expressions and solving polynomial equations. Consider the polynomial \(x^2 + 5x + 6\). We can factor it by finding two numbers that multiply to the constant term (6) and add up to the coefficient of the middle term (5). These numbers are 2 and 3, so\[x^2 + 5x + 6 = (x + 2)(x + 3)\]When factoring a polynomial, it is often helpful to:
- Find the GCF (Greatest Common Factor): If possible, use the GCF to simplify the polynomial first.
- Look for recognizable patterns: Such as difference of squares or perfect square trinomials.
- Trial and error: Especially for quadratics, this can involve testing factor pairs of the constant term.
Common Denominator
Finding a common denominator is crucial when solving fractional equations, as it allows you to combine fractions by aligning the denominators. A common denominator is a shared multiple of the denominators in each fraction. In the given problem, we had denominators \((x+12)\), \((x+3)\) and \((x^2 + 5x + 6)\), which factorizes to \((x+3)(x+2)\). Thus, the common denominator is \((x+12)(x+3)(x+2)\).Steps to find a common denominator:
- Factor all denominators: Break down each denominator into its simplest factors.
- Identify common factors: If multiple fractions share factors, use each factor the greatest number of times it appears.
- Multiply remaining factors: Multiply the remaining unique factors to complete the common denominator.
Discriminant Calculation
The discriminant in a quadratic equation is part of the quadratic formula and is represented by \(b^2 - 4ac\). It gives insight into the nature of the roots of the quadratic equation. Here鈥檚 how the discriminant guides our understanding:
- If \(b^2 - 4ac > 0\): The quadratic equation has two distinct real roots.
- If \(b^2 - 4ac = 0\): There is exactly one real root, or the roots are repeated (the parabola touches the x-axis at one point).
- If \(b^2 - 4ac < 0\): The roots are complex or imaginary, as the parabola does not intersect the x-axis.
