Problem 40 - Beginning and Intermediate Algebra

Question

Solve each equation. $$104 r+36=12 r^{2}$$

Step-by-Step Solution

Answer

The solutions for the equation \(104r + 36 = 12r^2\) are \(r = 9\) and \(r = -\frac{1}{3}\).

Step 1: Rewrite the equation in standard form

A quadratic equation in standard form looks like this: \(ax^2 + bx + c = 0\). Currently, the equation is given as \(104r + 36 = 12r^2\). To put the equation in standard form, we will move all terms to one side of the equation by subtracting \(104r + 36\) from both sides: \[12r^2 - 104r - 36 = 0\]

Step 2: Determine the solution method

Now that the equation is in standard form, we can determine if it can be solved by factoring or by using the Quadratic Formula. The Quadratic Formula is generally easier to use when the equation is not easily factorable. In this case, the equation \(12r^2 - 104r - 36 = 0\) does not appear to be easily factorable. We will proceed with using the Quadratic Formula instead, which is given by: \[r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Step 3: Solve for r using the Quadratic Formula

Applying the Quadratic Formula to our equation, we have the following values: a = 12, b = -104, and c = -36. Inserting these values, we get: \[r = \frac{-(-104) \pm \sqrt{(-104)^2 - 4(12)(-36)}}{2(12)}\] Simplifying the equation, \[r = \frac{104 \pm \sqrt{10816 + 1728}}{24}\] \[r = \frac{104 \pm \sqrt{12544}}{24}\] \[r = \frac{104 \pm 112}{24}\] Now, we have two possible solutions for r: \[r_1 = \frac{104 + 112}{24} = \frac{216}{24} = 9\] and \[r_2 = \frac{104 - 112}{24} = \frac{-8}{24} = -\frac{1}{3}\] Thus, the solution to the equation \(104r + 36 = 12r^2\) is: \[r = 9, -\frac{1}{3}\]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations that cannot be easily factored. Quadratic equations often take the form of \( ax^2 + bx + c = 0 \). When such equations don't have obvious factors, the quadratic formula provides a method for finding solutions. It is expressed as:\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Where:

• \( a \) is the coefficient of \( r^2 \)
• \( b \) is the coefficient of \( r \)
• \( c \) is the constant term

The term inside the square root, \( b^2 - 4ac \), is known as the discriminant. The discriminant tells us about the nature of the roots:

• If it's positive, there are two real and distinct solutions.
• If zero, there is one real repeated solution.
• If negative, the solutions are complex or imaginary.

To apply the quadratic formula effectively, ensure the equation is correctly set in standard form. Then, simply substitute the numerical values of \( a \), \( b \), and \( c \) into the formula, and solve for the variable.

Factoring

Factoring is a technique used to solve quadratic equations by expressing them as the product of two binomials. The equation is set to zero, and each factor is set equal to zero to find solutions. Factoring works best when the quadratic equation is simple or when the solution can be easily spotted. To factor a quadratic equation like \( ax^2 + bx + c \), you often look for two numbers that multiply to \( ac \) and add up to \( b \). Sometimes, it's straightforward, but not all equations are factorable with simple numbers. Here’s what you consider when trying to factor:

• Check if there is a common factor among the terms. If there is, factor it out first.
• Use special factoring formulas for perfect squares or differences of squares.
• Try different number combinations if necessary.

In many cases, if factoring proves too complex, it is often easier to rely on the quadratic formula instead. Understanding when to use each method is vital for efficiency in algebra.

Standard Form

The standard form of a quadratic equation is instrumental for analysis and solving purposes. A quadratic equation is in standard form if it is expressed as \( ax^2 + bx + c = 0 \). Correctly placing an equation in this form sets up subsequent steps needed for solving.Here’s the basic process of rearranging an equation into standard form:

• Ensure all terms are on one side of the equation, leaving zero on the other.
• Identify and reorganize the terms so that \( ax^2 \) is first, followed by \( bx \), and then the constant \( c \).
• Simplify any like terms or expression if needed.

Transforming into standard form ensures that you can consistently apply methods like factoring or the quadratic formula. It offers a clear path towards solving, as it levels different equations into a unified format.

Algebraic Solutions

Algebraic solutions refer to the various methods we apply to find solutions to equations, especially for quadratic equations. Among these methods, the most common are factoring and the quadratic formula. Each method has its own optimal conditions for use:

• Factoring is optimal when the numbers in the quadratic equation are manageable and make simple arithmetic sense.
• The quadratic formula is versatile, as it can be used when the equation doesn’t factor easily or if an exact solution is needed quickly.

Deciding which approach to use depends on a quick evaluation of the equation's complexity. Developing familiarity with each method enhances one's ability to approach quadratic equations methodically. Often, these methods work together:

• Begin with standard form.
• Attempt factoring to see if a straightforward solution is present.
• If not viable, apply the quadratic formula for a sure solution.

In algebra, these foundational techniques build the pathway to understanding more complex mathematical challenges.