Problem 52 - 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. $$y(y+9)=4(2 y+5)$$

Step-by-Step Solution

Answer

The solution to the equation is \(y=-5\) and \(y=4\).

Step 1: Simplify the Equation

First, simplify each side of the equation. \[y^2+9y = 4(2y+5)\] becomes \[y^2+9y = 8y+20\].

Step 2: Set the Equation to Zero

Second, isolate all terms on one side of the equation and set it to zero: \[y^2+9y - 8y - 20 = 0\] to get \[y^2+y - 20 = 0\].

Step 3: Factorize the Equation

Third, factorize the quadratic equation into binomials: \[(y+5)(y-4)=0\]

Step 4: Solve for the Variable

Next, equate each factor to zero and solve for \(y\): \[y+5=0 \implies y=-5\] and \[y-4=0 \implies y=4\]

Step 5: Check the Answers

Lastly, substitute \(y=-5\) and \(y=4\) into the original equation and verify if left side equals right side. If it does, then your solutions are correct.

Key Concepts

Factoring

Factoring is a method used to simplify algebraic expressions. It's especially handy for solving quadratic equations. A quadratic equation typically looks like this: \(ax^2 + bx + c = 0\). The idea is to rewrite this equation as a product of binomials.

• A binomial is an algebraic expression that contains two terms. For instance, \( (y+5) \) and \( (y-4) \) are binomials.
• The goal in factoring is to express the quadratic as a multiplication of two binomials.

When we multiply the binomials back (using the distributive property), we should get back the original quadratic equation.In the exercise given, we started with \(y^2 + y - 20 = 0\) and factored it into \((y+5)(y-4) = 0\). This is a critical step because it allows us to identify potential solutions to the equation. It's important to note that not every quadratic equation can be easily factored using integers. However, the factoring method is a quick and effective technique when applicable.

Solving Quadratic Equations

When solving quadratic equations by factoring, we take advantage of the zero product property. This property states that if the product of two values is zero, at least one of the values must be zero.Steps to solve by factoring:

• First, factor the equation, as we did by obtaining \((y+5)(y-4) = 0\).
• Next, set each binomial equal to zero and solve for the variable:
  • \(y + 5 = 0\) yields \(y = -5\).
  • \(y - 4 = 0\) yields \(y = 4\).

This process gives us the solutions of the quadratic equation. Each solution represents a point where the graph of the equation intersects the x-axis. By solving the equation, we find these intersection points, which are also known as roots or x-intercepts of the equation's graph.

X-Intercepts

X-intercepts of a quadratic function are the points where the graph of the equation crosses the x-axis. For each x-intercept, the y-value is zero. Identifying x-intercepts is crucial because they represent the solutions of the quadratic equation.

• In our example, the solutions \(y = -5\) and \(y = 4\) indicate x-intercepts at these values.
• Graphically, these are the points \((-5, 0)\) and \((4, 0)\) where the parabola crosses the x-axis.

Finding x-intercepts can be done algebraically, as we did by solving the equation \((y+5)(y-4) = 0\), or visually by plotting the function on a graph.The importance of finding x-intercepts in a quadratic equation is that they help us understand the roots and behavior of the quadratic function. Each x-intercept provides valuable insight into the function's graph; indicating where it touches or crosses the x-axis. Understanding this concept allows you to solve and graph quadratic equations more effectively.