Problem 17 - 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: Write Equation in Standard Form

First, we need to rearrange the given equation into the standard form of a quadratic equation, which is \(ax^2 + bx + c = 0\). The given equation is \(R^2 + 12 = 7R\).Subtract \(7R\) from both sides to obtain \(R^2 - 7R + 12 = 0\).

Step 2: Identify Coefficients for Factoring

In the equation \(R^2 - 7R + 12 = 0\), identify the coefficients. Here, \(a = 1\), \(b = -7\), and \(c = 12\). We need to find two numbers that multiply to \(c\) (12) and add up to \(b\) (-7).

Step 3: Find Factors of the Constant Term

List factors of the constant term \(c = 12\). Possible factor pairs are (1, 12), (2, 6), and (3, 4). We are looking for a pair that adds up to \(-7\).

Step 4: Choose the Correct Factor Pair

Among the factor pairs, (3, 4) is a good candidate. By making both factors negative, we have (-3, -4), which add up to \(-7\). Check by multiplying: \((-3) \times (-4) = 12\) and \((-3) + (-4) = -7\).

Step 5: Rewrite the Middle Term and Factor Completely

Rewrite the middle term, \(-7R\), using the pair found: \(-3R - 4R\). The equation becomes \(R^2 - 3R - 4R + 12 = 0\). Group terms: \((R^2 - 3R) + (-4R + 12)\). Factor each group: \(R(R - 3) - 4(R - 3) = 0\).

Step 6: Factor by Grouping

Notice \((R - 3)\) is common. Factor it out to get: \((R - 3)(R - 4) = 0\).

Step 7: Solve for Roots

Set each factor equal to zero and solve for \(R\): 1. \(R - 3 = 0\) gives \(R = 3\).2. \(R - 4 = 0\) gives \(R = 4\). Therefore, the solutions are \(R = 3\) and \(R = 4\).

Key Concepts

Factoring

Factoring is a method used to solve quadratic equations. It involves expressing the quadratic equation as a product of two expressions. When the product of these expressions equals zero, it is possible to use the zero-product property, which states that if the product of two factors is zero, at least one of the factors must be zero. Therefore, setting each factor equal to zero helps find the solutions or roots of the equation.

To factor a quadratic equation, follow these steps:

• Begin by identifying the equation in the standard form: \(ax^2 + bx + c = 0\).
• Next, find two numbers that multiply together to give the constant term \(c\) and add up to the coefficient of the linear term \(b\).
• Use these two numbers to rewrite the middle term of the equation.
• Finally, group terms and extract common factors to set up the equation as a product of two binomials.

By following these steps, the equation becomes factored and ready for solving.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is fundamental in understanding and solving quadratic equations. This form is written as \(ax^2 + bx + c = 0\).

Here's how to identify each part of the standard form:

• \(a\) is the coefficient of the quadratic term \(x^2\).
• \(b\) is the coefficient of the linear term \(x\).
• \(c\) is the constant term.

Standard form is crucial because it allows us to quickly determine the method for solving the equation, whether through factoring, completing the square, or using the quadratic formula. Simplifying an equation into its standard form often involves moving all terms to one side of the equation, resulting in the equation set to zero. This aligns with the zero-product property essential for factoring.

In the example equation \(R^2 + 12 = 7R\), subtracting \(7R\) from both sides moves all terms to one side, resulting in the standard form \(R^2 - 7R + 12 = 0\). This standardization is the first critical step in solving quadratic equations effectively.

Roots of Quadratic Equation

The roots of a quadratic equation are the solutions to the equation. These are the values of \(x\) (or the variable in use) where the quadratic function equals zero. In simpler terms, the roots are the points where the graph of the quadratic equation intersects the x-axis.

Finding the roots is the ultimate goal when solving quadratic equations. Here is how it is typically done:

• Once the quadratic equation is in standard form, factor it as necessary.
• Set each factor equal to zero. This is because of the zero-product property. If the product is zero, one or both factors must be zero.
• Solve the resulting simple equations.

For example, in the factored equation \((R - 3)(R - 4) = 0\), set \(R - 3 = 0\) and \(R - 4 = 0\). Solving these gives the roots \(R = 3\) and \(R = 4\). These roots mean that if you plug either back into the original equation \(R^2 + 12 = 7R\), the equation holds true or equals zero.

Understanding roots and how to find them is essential because they provide crucial insights into the behavior of the quadratic function, indicating where the curve meets or crosses the x-axis.