Problem 38 - Algebra 1: Concepts and Skills

Question

Factor first, then solve the equation. Check your solutions. \(\frac{1}{y^{2}-16}-\frac{2}{y+4}=\frac{2}{y-4}\)

Step-by-Step Solution

Answer

The solutions to the equation are \(y=14\) and \(y=1\).

Step 1: Combine Fractions

To combine the two fractions on the left side of the equation, we need a common denominator. The denominators are \(y^{2}-16\) and \(y+4\). We notice that \(y^{2}−16\) is a difference of squares which factors to \((y+4)(y-4)\). So, we multiply the second term with \((y-4)\) to get the common denominator:\(\frac{1}{y^{2}-16}-\frac{2(y-4)}{(y+4)(y-4)}=\frac{2}{y-4}.\) Now we can combine the terms on the left side of the equation: \(\frac{1-2y+8}{(y+4)(y-4)} = \frac{2}{y-4}\). This simplifies to: \(\frac{9-2y}{(y+4)(y-4)} = \frac{2}{y-4}.\)

Step 2: Cross-Multiply

Cross-multiplying to eliminate the denominators gives: \((9-2y)*(y-4) = (y+4)*2\). This simplifies to \(2y^{2} -36y +36 = 2y+8\).

Step 3: Simplify and Solve the Quadratic Equation

Subtract 2y and 8 from both sides to put the equation in the standard form \(a*x^{2} + b*x + c = 0\): \(2y^{2} - 38y +28 =0. \)Then, factoring the equation to its factors, gives: \(2*(y-14)*(y-1)=0. \)Setting each factor equal to zero gives the solutions: \(y=14, y=1.\)

Key Concepts

Factoring

Factoring is a key algebraic method used to break down complex expressions into more manageable parts. This can make equations simpler to solve. In this exercise, the expression \(y^2 - 16\) represents a difference of squares. This type of expression is special because it can be rewritten as the product of two binomials:

• The square root of the first term: \(y\)
• The square root of the second term: \(4\)

Thus, \(y^2 - 16\) can be factored into \((y + 4)(y - 4)\).
By expressing complex expressions as products of simpler terms, factoring simplifies calculations dramatically.
This process is crucial to finding a common denominator in rational expressions.

Rational Expressions

Rational expressions involve fractions with polynomials in the numerator, the denominator, or both. Solving these types of expressions often starts by finding a common denominator.
In the given exercise, we start with two fractions:

• \(\frac{1}{y^2 - 16}\)
• \(\frac{2}{y + 4}\)

To combine these, we use the factored form \((y + 4)(y - 4)\) as the common denominator. Multiplying the necessary terms gives each fraction a common base, allowing us to combine them into one.
After finding the common denominator and combining the fractions, the rational expression can be further simplified and solved.
Handling rational expressions carefully ensures accurate solutions to equations involving fractions.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, generally in the form \(ax^2 + bx + c = 0\). Solving these equations involves finding the values of \(x\) (or \(y\) in our case) that satisfy the equation.In the exercise, cross-multiplying the simplified expressions leads to a quadratic equation: \(2y^2 - 36y + 36 = 2y + 8\).

• Simplify the equation by subtracting \(2y\) and \(8\) from both sides, resulting in \(2y^2 - 38y + 28 = 0\).
• Next, factor the quadratic into \(2(y - 14)(y - 1) = 0\).

Setting each factor equal to zero helps find the solutions: \(y = 14\) and \(y = 1\).
Once the quadratic is factored into factors, solving it often involves simply setting these factors to zero and solving for the variable.
Understanding this process is fundamental to solving more complex algebraic equations.