Problem 15 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation by factoring, if required: $$ z(2 z+1)=6 $$

Step-by-Step Solution

Answer

The solutions for the given equation are \(z = -4\) and \(z = \frac{3}{2}\).

Step 1: Expand the expression on the left side

First, we need to expand the expression on the left side of the equation by multiplication. We have the equation: \(z(2z + 1) = 6\) To expand this, multiply z by each term inside the parenthesis: \(2z^2 + z = 6\)

Step 2: Rearrange the equation to get a quadratic equation

Next, we want to get this equation in the standard form, which is: \(ax^2 + bx + c = 0\) To do that, let's subtract 6 from both sides: \(2z^2 + z - 6 = 0\) Now the equation is in standard form and can be factored if possible.

Step 3: Factor the quadratic equation

To factor the quadratic equation, we look for two numbers that multiply to the product of the first and last terms (2 * -6 = -12), and add up to the middle coefficient, which is 1. We find that the numbers 4 and -3 fit this. So, the factored form of the equation is: \((z + 4)(2z - 3) = 0\)

Step 4: Find the solutions

Now that we have the factored form, we can set each factor equal to zero and solve for z: 1) \(z + 4 = 0\): Subtract 4 from both sides: \(z_1 = -4\) 2) \(2z - 3 = 0\): Add 3 to both sides: \(2z = 3\) Divide by 2: \(z_2 = \frac{3}{2}\) So, the solutions for the given equation are \(z = -4\) and \(z = \frac{3}{2}\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of degree 2. This means that the highest power of the variable is squared, or raised to the second power. The general form of a quadratic equation is written as:\[ ax^2 + bx + c = 0 \]where:

• \( a \) is the coefficient of \( x^2 \), which must not be zero.
• \( b \) is the coefficient of \( x \).
• \( c \) is the constant term.

The equation can be solved using various methods, such as factoring, completing the square, or using the quadratic formula. The key characteristic of quadratic equations is that they typically produce a parabolic graph, which can open upward or downward depending on the sign of \( a \). Understanding the properties of quadratic equations is crucial for solving them efficiently.

Standard Form

Putting a quadratic equation in standard form is an essential step in solving it. The standard form is \( ax^2 + bx + c = 0 \), where all terms are on one side of the equation, set equal to zero.To achieve this, you must rearrange and simplify the equation. This usually involves expanding expressions and combining like terms, as illustrated in the example:- Start with the equation \( z(2z + 1) = 6 \).- Expand the left-hand side: \( 2z^2 + z = 6 \).- Subtract 6 from both sides to obtain: \( 2z^2 + z - 6 = 0 \).A neat and clear form is crucial for further steps like factoring, as it lays the groundwork for identifying patterns and relationships between the coefficients.

Roots of Equation

The roots of a quadratic equation are the values of the variable that make the equation true. These roots are also commonly referred to as solutions or zeros of the equation.We can find these roots by factoring, completing the square, or using the quadratic formula. In the context of the given problem, we found the roots by factoring. Once the equation \((z + 4)(2z - 3) = 0\) is established, the solutions are identified by setting each factor equal to zero.- Setting \( z + 4 = 0 \) results in the solution \( z_1 = -4 \).- Setting \( 2z - 3 = 0 \), after solving for \( z \), results in the root \( z_2 = \frac{3}{2} \).These solutions indicate the points at which the parabolic graph of the quadratic crosses the \( z \)-axis.

Factoring Method

Factoring is a powerful method for solving quadratic equations because it can simplify the process if performed correctly. This method involves rewriting the quadratic expression as a product of two binomials.To factor a quadratic equation effectively:- Identify two numbers that multiply to the product of the coefficient of the squared term and the constant term (\( a \cdot c \)) and add up to the coefficient of the linear term (\( b \)).- For the equation \( 2z^2 + z - 6 = 0 \), multiply \( 2 \) and \( -6 \) to get \(-12\). The numbers \( 4 \) and \( -3 \) add up to \( 1 \) (the coefficient of \( z \)) and multiply to \(-12\).- Rewrite and factor the quadratic: \((z + 4)(2z - 3) = 0\).The factoring method not only helps in solving the equation but also provides a clear insight into the properties and nature of the solutions.