Problem 51 - Introductory Algebra for College Students

Question

Use factoring to solve each quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\) -intercepts. $$y(y+8)=16(y-1)$$

Step-by-Step Solution

Answer

The solution is \( y = 4 \).

Step 1: Rewrite the equation

Let's start by opening out the brackets and bringing everything onto one side of the equal sign to get a quadratic equation.\( y^2 + 8y = 16y - 16 \) simplifies to \( y^2 - 8y + 16 = 0 \)

Step 2: Factoring the quadratic equation

The equation \( y^2 - 8y + 16 = 0 \) can be factored to \( (y-4)^2 = 0 \) because the equation is a perfect square trinomial.

Step 3: Solving for y

Setting \( (y-4)^2 = 0 \) gives us \( y = 4 \).

Step 4: Verification

Substitute \( y = 4 \) back into the original equation to verify. \( 4(4+8) = 16(4-1) \), which simplifies to \( 48 = 48 \), thus confirming that the solution is valid.

Step 5: Graphical verification

Another way to verify the solution is by plotting \( y(y+8) \) and \( 16(y-1) \) in a graphing utility and looking for the x-intercept. The x-intercept will be equal to the y-value obtained by solving the equation algebraically. In this case, the graph of both equations will intersect at \( y = 4 \) demonstrating that the solution is correct.

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two. It typically takes the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. Quadratic equations form a parabola when graphed, which can open upwards or downwards depending on the sign of \( a \).

Understanding how to rewrite an equation into the standard quadratic form is essential in solving or factoring it. In this exercise, we start from the equation \( y(y+8) = 16(y-1) \). By expanding and rearranging it, we get \( y^2 - 8y + 16 = 0 \).

This transformation showcases a methodical approach to handle quadratic equations:

• Expand all expressions fully.
• Move all terms to one side of the equation.
• Rearrange them to match the quadratic form.

Working with quadratic equations often involves various techniques such as factoring, completing the square, and using the quadratic formula. Each of these methods helps find solutions for \( x \) that make the equation true.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be factored into an identical binomial squared. It follows the pattern \( (x + b)^2 = x^2 + 2bx + b^2 \). Recognizing a perfect square trinomial allows for quick and efficient factoring.

In our case, the equation \( y^2 - 8y + 16 = 0 \) is a perfect square trinomial. This equation can be expressed as \( (y-4)^2 = 0 \). When the trinomial is factored into \( (y-4)^2 \), it shows us that the solution for \( y \) is found by solving \((y-4) = 0 \), which gives \( y = 4 \).

The beauty of perfect square trinomials lies in their symmetrical nature, making them straightforward to factor. Recognizing them allows you to not only solve the equation quickly but also to reduce potential mistakes in the calculation process.

Graphical Verification

Graphical verification is a valuable technique to confirm solutions obtained algebraically. This involves graphing the functions described by the equation and checking the visual intercepts or intersection points.

According to the original exercise, after solving \( y^2 - 8y + 16 = 0 \) and obtaining \( y = 4 \) algebraically, we verify graphically. By plotting both \( y(y+8) \) and \( 16(y-1) \) on a graphing utility, the intersection of their graphs provides confirmation of the solution.

This step assures that both algebraic and graphical solutions coincide, offering a deeper understanding of the problem. Graphical analysis is especially useful when dealing with complex problems where visualization provides insights. It acts as a double-check mechanism to safeguard against possible algebraic errors.

• Plot both expressions independently.
• Identify the intersection point.
• Ensure it matches the algebraic solution.

By adopting both algebraic and graphical methods, you enhance your problem-solving toolkit, making sure your solutions are sound and well-justified.