Problem 28 - A Graphical Approach to College Algebra

Question

Solve each equation. For equations with real solutions, support your answers graphically. $$5 x^{2}+3 x=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = -\frac{3}{5}\).

Step 1: Factor Out the Common Term

Start by identifying the common factor in the equation. The equation given is \(5x^2 + 3x = 0\). Both terms have a common factor of \(x\). Factoring \(x\) out of the equation, we get: \[ x(5x + 3) = 0 \]

Step 2: Apply the Zero Product Property

According to the zero product property, if the product of two numbers is zero, at least one of the numbers must be zero. This gives us \[ x = 0 \] and \[ 5x + 3 = 0 \].

Step 3: Solve for x in Each Equation

First, solve the linear equation \(x = 0\), which is already solved. Next, solve \(5x + 3 = 0\) by isolating \(x\):1. Subtract 3 from both sides: \[ 5x = -3 \]2. Divide both sides by 5: \[ x = -\frac{3}{5} \]

Step 4: Verify the Solutions Graphically

Graph the function \(y = 5x^2 + 3x\). The solutions to the equation correspond to the x-intercepts of this graph. The graph is a parabola opening upwards. It crosses the x-axis at \(x = 0\) and \(x = -\frac{3}{5}\). These intercepts confirm that the solutions to the equation are \(x = 0\) and \(x = -\frac{3}{5}\).

Key Concepts

Zero Product Property

The Zero Product Property is an essential concept when solving quadratic equations. It states that if the product of two numbers is zero, then at least one of the numbers must be zero. Suppose we have an equation like \( a \cdot b = 0 \). According to this property, either \( a = 0 \), \( b = 0 \), or both.
When applying this to quadratic equations, it often involves factoring the equation into a product of two simpler expressions. For example, consider the equation \( x(5x + 3) = 0 \). Here, we have factored the quadratic equation into two parts, \( x \) and \( 5x + 3 \).

• If \( x = 0 \), then the equation holds true.
• If \( 5x + 3 = 0 \), solving for \( x \) gives us another solution.

This property helps break down complex equations into more manageable parts, making it easier to find all possible solutions for \( x \).

Factoring Quadratic Equations

Factoring is a technique used to express a quadratic equation as the product of its factors. The goal of factoring is to rewrite the quadratic equation in a simplified form, typically as a product of two binomial expressions. This often makes it easier to solve the equation.
For the equation \( 5x^2 + 3x = 0 \), we first look for any common factors in all terms. In this case, both terms contain an \( x \). By factoring out an \( x \), the equation becomes \( x(5x + 3) = 0 \).
This process simplifies the equation and allows us to apply the Zero Product Property. Factoring not only reduces the complexity of the problem but also reveals the solutions more clearly.

• Identify common factors in each term.
• Factor out the common term.
• Rewrite the equation in its factored form.

Once factored, it paints a clearer picture of the solutions: either \( x = 0 \) or solve \( 5x + 3 = 0 \) to find another solution for \( x \).

Graphical Solution of Quadratics

Graphing is a powerful visual tool to verify solutions of a quadratic equation. It helps us understand the behavior and solutions of the equation by showing where the graph crosses the x-axis, known as x-intercepts. For the equation \( y = 5x^2 + 3x \), plotting the graph reveals a parabola that opens upwards.
To find out where this parabola crosses the x-axis, we look for points where \( y = 0 \), which corresponds to the original equation \( 5x^2 + 3x = 0 \). These intersections are exactly the solutions found algebraically. For this exercise, the graph confirms the solutions \( x = 0 \) and \( x = -\frac{3}{5} \).
Graphical verification is especially useful because:

• It visually confirms algebraic solutions.
• It helps detect and correct any errors in calculations.
• It provides insight into the nature of the quadratic function, such as the direction of its opening and its vertex.

Therefore, combining algebraic and graphical methods provides a comprehensive approach to verifying solutions and understanding quadratic equations.