Problem 80 - Algebra 2

Question

Solve each equation. Check your solutions. \(\frac{2 a-5}{a-9}-\frac{a-3}{3 a+2}=\frac{5}{3 a^{2}-25 a-18}\)

Step-by-Step Solution

Answer

The solutions are \(a = 2.8\) and \(a = -3\).

Step 1: Identify the Common Denominator

To solve the given rational equation \(\frac{2a-5}{a-9} - \frac{a-3}{3a+2} = \frac{5}{3a^2 - 25a - 18}\), start by finding a common denominator for all three terms. The common denominator here is \((a-9)(3a+2)\), which is the product of the individual denominators.

Step 2: Rewrite Each Term with the Common Denominator

Rewrite each term so that they have the same denominator:\[\frac{(2a-5)(3a+2)}{(a-9)(3a+2)} - \frac{(a-3)(a-9)}{(a-9)(3a+2)} = \frac{5}{(3a^2 - 25a - 18)}\]Express \(\frac{5}{3a^2 - 25a - 18}\) in terms of the common denominator as \(\frac{5}{(a-9)(3a+2)}\).

Step 3: Combine the Left Side of the Equation

Combine the fractions on the left-hand side:\[\frac{(2a-5)(3a+2) - (a-3)(a-9)}{(a-9)(3a+2)}\]Now, simplify the numerator by expanding the product of polynomials.

Step 4: Expand and Simplify the Numerator

Expand and simplify the expressions within the numerators:For \((2a-5)(3a+2)\):\(6a^2 + 4a - 15a - 10 = 6a^2 - 11a - 10\)For \((a-3)(a-9)\):\(a^2 - 9a - 3a + 27 = a^2 - 12a + 27\)Thus, the expression inside the numerator becomes:\(6a^2 - 11a - 10 - (a^2 - 12a + 27) = 5a^2 + a - 37\)

Step 5: Set the Simplified Left-Side Equal

The left side of the equation is now:\[\frac{5a^2 + a - 37}{(a-9)(3a+2)} = \frac{5}{(a-9)(3a+2)}\]With the same denominators on both sides, we can equate the numerators:\[5a^2 + a - 37 = 5\]

Step 6: Solve the Quadratic Equation

Move all terms to one side to form the quadratic equation:\[5a^2 + a - 42 = 0\]Solve this equation using the quadratic formula:\[a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Where \(a = 5\), \(b = 1\), and \(c = -42\).Calculate the discriminant \(b^2 - 4ac = 1 + 840 = 841\) and solve:\[a = \frac{-1 \pm \sqrt{841}}{10}\]\[a = \frac{-1 \pm 29}{10}\]

Step 7: Calculate Possible Solutions

Compute the values:1. \(a = \frac{28}{10} = 2.8\)2. \(a = \frac{-30}{10} = -3.0\)

Step 8: Check the Solution

Substitute both solutions back into the original equation to verify:For \(a = 2.8\) and \(a = -3\), ensure they don't make any denominator zero and result in the original equation holding true.You will find that both solutions satisfy the original equation.

Key Concepts

Common Denominator

In rational equations, finding a common denominator is a crucial first step. It allows us to combine multiple rational expressions into a single expression, which simplifies solving the equation. A common denominator is essentially the least common multiple (LCM) of all the denominators involved in the equation.
To find a common denominator:

• Identify each distinct denominator in your equation.
• For each denominator, consider if there are any factors shared with another denominator.
• The common denominator will be the product of the unique factors from each separate denominator.

In our exercise, the rational equation involves the denominators \(a-9\) and \(3a+2\). Therefore, the common denominator is the product of these two, \((a-9)(3a+2)\). This common denominator allows each term to be rewritten with the same base. This step simplifies the process of evaluating and manipulating the rational expressions in the subsequent steps.

Quadratic Formula

The quadratic formula is a powerful tool in algebra for solving second-degree polynomial equations, which have the general form \(ax^2 + bx + c = 0\). The quadratic formula states that the solutions for \(x\) can be found using:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]The variables \(a\), \(b\), and \(c\) are the coefficients of the terms in the quadratic equation. A vital part of this formula is the discriminant: \(b^2 - 4ac\). The discriminant determines the nature of the roots:

• If it is positive, there are two distinct real roots.
• If zero, there is exactly one real root (repeated root).
• If negative, the roots are complex and not real numbers.

In our exercise, after rewriting the rational equation with a common denominator, we ended up with a quadratic equation \(5a^2 + a - 42 = 0\) to solve. We use the quadratic formula by plugging in \(a = 5\), \(b = 1\), and \(c = -42\) to find the solutions for \(a\). Calculating gives us the real roots: \(a = 2.8\) and \(a = -3\).

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. Solving problems involving rational expressions requires careful attention, especially to ensure that we are not performing illegal operations like dividing by zero.
Key steps in solving equations involving rational expressions include:

• Determine any excluded values for the variable that make the denominator zero. This is crucial before performing any algebraic manipulation to prevent undefined expressions.
• Use a common denominator to combine separate rational expressions as one.
• Simplify the resulting expression and solve for the unknown variable by treating the expression as an algebraic equation.

In the problem at hand, initial denominators \(a-9\) and \(3a+2\) must be considered, as substituting specific values for \(a\) that make either denominator zero would invalidate the expressions. This is why the solutions should be checked to ensure they do not cause division by zero.

Numerator Simplification

Simplifying the numerator is an essential step when working with rational expressions. Once you have a common denominator, the next step often involves simplifying and reducing the expression in the numerators of the involved fractions. This process helps transform a complex equation into something more manageable.
Here's how you can approach numerator simplification in rational expressions:

• Expand any polynomials in the numerators by distributing any factors across terms within parentheses.
• Combine like terms by adding or subtracting coefficient values.
• Ensure that the expression within the numerator is completely reduced. Simplifying might also involve factoring back the expressions if possible to cancel out with any common factors in the denominator.

In our specific exercise, after rewriting with a common denominator, we simplified the numerator by expanding \((2a-5)(3a+2)\) and \((a-3)(a-9)\). The simplification yielded \(6a^2 - 11a - 10 - (a^2 - 12a + 27)\). This then simplifies further to \(5a^2 + a - 37\), allowing us to solve it as a simplified quadratic equation.