Problem 25 - 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. $$3 x^{2}=-5 x$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 0\) and \(x = -5/3\).

Step 1: Rearrange the Equation

Rearrange the quadratic equation by adding \(5x\) on both sides to make the equation equal to zero, resulting in \(3x^{2} + 5x = 0\).

Step 2: Factor Out the Common Factor

Factor out the common factor which is \(x\) from the quadratic expression, giving \(x(3x + 5) = 0\).

Step 3: Solve for \(x\)

Set each factor equal to zero and solve for \(x\) in order to find the solution(s). So, \(x = 0\) and \(3x + 5 = 0\) Finally, solving for \(x\) in the second equation gives \(x = -5/3\).

Step 4: Check the Solution

Substitute each solution back into the original equation to verify. Substituting \(x = 0\) results in \(3(0)^2 = -5(0)\), implying that \(0 = 0\), which holds true. Substituting \(x = -5/3\) results in \(3(-5/3)^2 = -5(-5/3)\), implying that \(5 = 5\), which is also true.

Key Concepts

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants with \(a eq 0\), and \(x\) represents the variable. The equation is called 'quadratic' because 'quad' pertains to the square. This means the highest power of \(x\) is 2, resulting in a parabolic graph. This makes quadratic equations unique and versatile in capturing relationships with variables that grow exponentially. In our exercise, the given equation to solve is \(3x^{2} + 5x = 0\). The challenge lies in finding the values of \(x\) that satisfy the equation, known as roots. To solve these equations effectively, various methods like factoring, completing the square, and using the quadratic formula are commonly employed.

Factoring Method

The factoring method is a straightforward approach to solve quadratic equations. It involves expressing the equation in a product of two simpler expressions set to zero. The crux of this method relies on the Zero-Product Property. This property states if the product of two factors equals zero, then at least one of the factors must be zero. In our solution, after rearranging the given equation \(3x^2 + 5x = 0\), we factored out the common term \(x\), achieving \(x(3x + 5) = 0\). By isolating the \(x\), we simplify the process of finding solutions. Factoring is often preferred due to its simplicity, particularly when the equation coefficients are integers, making mental calculations more manageable.

Solving Equations

Once the equation is factored, the next step is to solve the resulting simpler equations. Remember, solving each factor individually is key. In the example from the exercise, after factoring, we set each factor equal to zero:

• \(x = 0\)
• \(3x + 5 = 0\)

The first factor provides a direct solution: \(x = 0\). For the second factor \(3x + 5 = 0\), we perform basic algebra to isolate \(x\): subtract 5 from both sides, then divide by 3, resulting in \(x = -\frac{5}{3}\). Both values are solutions to the original quadratic equation. Solving the equations derived from factors requires careful handling of algebraic operations to avoid errors and ensure accuracy.

X-Intercepts

In the context of quadratic equations, \(x\)-intercepts are the points where the graph of the equation crosses the \(x\)-axis. These intercepts correspond to the solutions or roots of the equation. For the quadratic equation given in our exercise, the \(x\)-intercepts can be seen as the solutions \(x = 0\) and \(x = -\frac{5}{3}\). Understanding \(x\)-intercepts is crucial because:

• They reveal where the function's value changes sign.
• Help in visualizing the roots.
• Confirm that our factored equation solutions are correct.

To verify this using a graphing utility, plot the equation \(y = 3x^2 + 5x\). The intercepts are clear where the curve meets the \(x\)-axis, at \(0\) and \(-\frac{5}{3}\), confirming our factored solution's accuracy.