Problem 35 - Elementary Algebra

Question

Solve the fractional equation \(\frac{4}{x+12}+\frac{3}{x+3}=\frac{4}{x^{2}+5 x+6} .\)

Step-by-Step Solution

Answer

Answer: The two solutions to the given fractional equation are x = -2 and x = -6.

Step 1: 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 \((x + 12)(x + 3)(x + 2)\).

Step 2: Multiply each term by the LCM

Now we will multiply each term of the equation by the LCM, to eliminate the denominators: \[(x + 12)(x + 3)(x + 2) \cdot \frac{4}{(x+12)} \,+\, (x + 12)(x + 3)(x + 2) \cdot \frac{3}{(x+3)} \,=\, (x + 12)(x + 3)(x + 2) \cdot \frac{4}{(x^2+5x+6)}\]

Step 3: 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: \[7x^2 + 56x + 72 = 4x + 48\]

Step 4: 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 \(x = -6\).

Key Concepts

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.

The quadratic formula is a powerful tool as it works for all quadratic equations, providing solutions by incorporating the discriminant to determine if roots are real or complex.

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.

Understanding how to factor polynomials allows for simplification of problems, especially when dealing with larger equations where direct calculation is complicated.

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.

This approach simplifies the original equation by converting each fraction into a version with the common denominator, allowing you to eliminate fractions by multiplying through by this 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’s 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.

In the exercise, calculating the discriminant (2304) revealed that the quadratic equation had two distinct real roots, enabling us to find meaningful solutions to the equation. Understanding the discriminant is key to determining the most efficient method to solve a quadratic equation.