Problem 15 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by factoring. $$R^{2}+12=7 R$$

Step-by-Step Solution

Answer

The solutions are \( R = 3 \) and \( R = 4 \).

Step 1: Arrange the Equation

First, rewrite the quadratic equation in the standard form by moving all terms to one side: \( R^2 + 12 = 7R \). Subtract \( 7R \) from both sides to get: \( R^2 - 7R + 12 = 0 \).

Step 2: Identify the Coefficients

For the quadratic equation \( R^2 - 7R + 12 = 0 \), identify the coefficients: \( a = 1 \), \( b = -7 \), and \( c = 12 \). These values will help in the factoring step.

Step 3: Factor the Quadratic

Find two numbers that multiply to \( c = 12 \) and add up to \( b = -7 \). These numbers are \(-3\) and \(-4\). Thus, the equation can be factored as \((R - 3)(R - 4) = 0\).

Step 4: Solve for R

Set each factor equal to zero: \( R - 3 = 0 \) or \( R - 4 = 0 \). Solving these gives \( R = 3 \) or \( R = 4 \). These are the solutions to the equation.

Key Concepts

Factoring

Factoring is an essential method for solving quadratic equations. It involves expressing the equation as a product of its factors. In the context of a quadratic equation, like \( R^2 - 7R + 12 = 0 \), we look for two numbers that multiply to the last constant term, \( c \), and add up to the middle coefficient, \( b \). This technique aids in simplifying the equation because once factored, it can be broken down into two simpler linear factors. For instance, in our example, the numbers \(-3\) and \(-4\) multiply to give \(12\) and sum up to \(-7\). Hence, the factored form of the equation is \((R - 3)(R - 4) = 0\). This strategic simplification allows us to solve the quadratic equation easily by setting each factor equal to zero. Factoring is a powerful tool for quickly solving quadratics and finding their roots.

Standard Form

Quadratic equations need to be in a specific format known as the standard form to apply various solving techniques accurately. The standard form of a quadratic equation is given by \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. To rearrange any quadratic equation into this form, one needs to bring all the terms to one side of the equation, ensuring the other side equals zero. In our exercise, the equation \( R^2 + 12 = 7R \) was rearranged into \( R^2 - 7R + 12 = 0 \) by subtracting \( 7R \) from both sides. Transitioning an equation into its standard form is crucial, as it lays the groundwork for identifying the coefficients and employing methods like factoring to find solutions.

Coefficients

Coefficients in a quadratic equation play a pivotal role during the solving process. They are the numerical values placed in front of variables or variable terms. In the standard form \( ax^2 + bx + c = 0 \), \( a \, (\text{for}\, x^2)\), \( b \, (\text{for}\, x)\), and \( c \, (\text{constant term})\) are the coefficients. From our quadratic example \( R^2 - 7R + 12 = 0 \), the coefficients are identified as follows: \( a = 1 \), \( b = -7 \), and \( c = 12 \). These coefficients are integral in determining the steps for factoring. They help identify the patterns or operations needed to simplify and solve the equation. Without correctly identifying the coefficients, approaching the solution becomes significantly more challenging.

Solving Equations

Solving quadratic equations involves finding the values of the variable that make the equation true. After factoring the equation, the subsequent step is to solve the factors separately. This is possible because, if the product of two terms equals zero, at least one of the terms must be zero itself. For the equation \( (R - 3)(R - 4) = 0 \), you set each factor to zero and solve each individually:

• \( R - 3 = 0 \) gives \( R = 3 \).
• \( R - 4 = 0 \) gives \( R = 4 \).

These solutions, \( R = 3 \) and \( R = 4 \), represent the roots of the original quadratic equation. Solving quadratic equations often involves performing operations step-by-step, as seen here, ensuring accuracy and comprehension. Understanding this process is crucial for successfully determining the solutions for any given quadratic equation.