Problem 43 - Algebra 2

Question

Solve each equation by using the method of your choice. Find exact solutions. \(10 x^{2}+3 x=0\)

Step-by-Step Solution

Answer

The solutions are \( x = 0 \) and \( x = -\frac{3}{10} \).

Step 1: Identify the equation type

The equation given is a quadratic equation of the form \( ax^2 + bx + c = 0 \). In our case, it is \( 10x^2 + 3x = 0 \), with \( a = 10 \), \( b = 3 \), and \( c = 0 \).

Step 2: Factor out the common term

Notice that both terms on the left-hand side of the equation, \( 10x^2 \) and \( 3x \), share a common factor of \( x \). Factor \( x \) out of the equation: \[ x(10x + 3) = 0 \].

Step 3: Solve each factor set to zero

Set each factor equal to zero and solve for \( x \). The equation \( x = 0 \) gives us one solution directly. For \( 10x + 3 = 0 \), solve for \( x \): \[ x = -\frac{3}{10} \].

Step 4: Verify the solutions

Substitute \( x = 0 \) and \( x = -\frac{3}{10} \) back into the original equation to ensure they satisfy it. Both values satisfy the equation, confirming they are correct solutions.

Key Concepts

Factoring Method

The factoring method is a popular technique for solving quadratic equations. It is particularly helpful when the quadratic can be transformed into a product of simpler expressions. In our problem, the equation given is a quadratic, identified by the form \( ax^2 + bx + c = 0 \). For this specific equation, \( 10x^2 + 3x = 0 \), we can see that both terms on the left side have a common factor, \( x \). The factoring method involves rewriting the equation as a product of its factors. Here, when we factor out \( x \), we obtain \( x(10x + 3) = 0 \). This is crucial because according to the zero-product property, if the product of two expressions equals zero, at least one of the expressions must be zero. By setting each factor equal to zero, we can break the equation into smaller, more manageable parts:

• \( x = 0 \)
• \( 10x + 3 = 0 \)

Solving \( 10x + 3 = 0 \) further gives the solution \( x = -\frac{3}{10} \). The factoring method simplifies the process of finding solutions by focusing on breaking down complex equations into parts that are easier to handle.

Solutions Verification

Verification is a key step in solving quadratic equations, as it ensures the accuracy of the solutions found. After solving a quadratic equation using the factoring method, it's important to substitute the solutions back into the original equation to confirm they are correct. Let’s take our equation \( 10x^2 + 3x = 0 \) as an example.When we solved the equation, we found the solutions \( x = 0 \) and \( x = -\frac{3}{10} \). Checking these solutions involves plugging them back into the original equation:

• For \( x = 0 \): Substitute \( x \) into \( 10x^2 + 3x = 0 \), which simplifies directly to \( 0 \), confirming our calculation.
• For \( x = -\frac{3}{10} \): Substitute \( x \), and calculate \( 10(-\frac{3}{10})^2 + 3(-\frac{3}{10}) \). This simplifies and also results in \( 0 \).

Both checks align with the original equation, verifying that \( x = 0 \) and \( x = -\frac{3}{10} \) are indeed correct solutions. Verification not only confirms the validity of the process but also enhances understanding by showing that the solutions satisfy the initial conditions.

Quadratic Equation Solving Methods

Quadratic equations, appearing in the form \( ax^2 + bx + c = 0 \), can be tackled using various methods, each suited to different kinds of quadratic scenarios. Here’s a brief overview of some common solving methods:

• Factoring: As demonstrated in our example, factoring is used when the equation can be expressed as a product of simple binomials. It works best when coefficients are integers and one can easily identify factors.
• Quadratic Formula: The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) is applicable universally for any quadratic equation. It is especially useful when factoring is difficult or impossible.
• Completing the Square: This method involves transforming the quadratic equation into a perfect square trinomial. It is helpful for deriving the quadratic formula and solving equations where the quadratic term's coefficient is not 1.
• Graphing: Although not algebraic, graphing can provide a visual solution, allowing one to see the points where the quadratic function intersects the x-axis. These intersection points are the solutions to the equation.

Each method has its strengths and applications, empowering you to choose the most convenient one depending on the complexity and form of the given quadratic equation. Understanding these methods enhances problem-solving flexibility and efficiency.