Problem 40 - Brief Applied Calculus

Question

Solve each equation by factoring or the Quadratic Formula, as appropriate. $$ 3 x^{2}+18=15 x $$

Step-by-Step Solution

Answer

The solutions are \( x = 3 \) and \( x = 2 \).

Step 1: Rewrite the Equation in Standard Form

First, we need to rewrite the equation so that all terms are on one side, forming a standard quadratic equation of the form \( ax^2 + bx + c = 0 \). Start with the given equation:\[ 3x^2 + 18 = 15x \]Subtract \( 15x \) from both sides to get:\[ 3x^2 - 15x + 18 = 0 \]

Step 2: Find the Discriminant

The discriminant of a quadratic equation is given by \( b^2 - 4ac \). For the equation \( 3x^2 - 15x + 18 = 0 \), identify \( a = 3 \), \( b = -15 \), and \( c = 18 \).Calculate the discriminant:\[ (-15)^2 - 4(3)(18) = 225 - 216 = 9 \]

Step 3: Apply the Quadratic Formula

Since the discriminant is positive, there are real solutions. Use the quadratic formula to find the solutions:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute the values \( a = 3 \), \( b = -15 \), and \( c = 18 \):\[ x = \frac{-(-15) \pm \sqrt{9}}{2 \times 3} \]This simplifies to:\[ x = \frac{15 \pm 3}{6} \]

Step 4: Simplify to Find the Solutions

Solve for the two possible values of \( x \):1. \( x = \frac{15 + 3}{6} = \frac{18}{6} = 3 \)2. \( x = \frac{15 - 3}{6} = \frac{12}{6} = 2 \)Thus, the solutions are \( x = 3 \) and \( x = 2 \).

Key Concepts

Quadratic Formula

The quadratic formula is a useful tool for solving quadratic equations, especially when they cannot be easily factored. This formula provides the solutions to any quadratic equation in the form \( ax^2 + bx + c = 0 \). The quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation. The symbol \( \pm \) indicates that there might be two solutions, depending on the value of the discriminant (\( b^2 - 4ac \)).To use this formula effectively:- Identify \( a \), \( b \), and \( c \) from the given quadratic equation.- Calculate the discriminant \( b^2 - 4ac \) to determine the number and type of solutions.- Substitute \( a \), \( b \), and \( c \) into the formula to solve for \( x \).The quadratic formula can provide both real and complex solutions, making it a versatile method when dealing with quadratic equations.

Discriminant

The discriminant of a quadratic equation plays a crucial role in understanding the nature of the equation's solutions. It is calculated using the expression \( b^2 - 4ac \) from the standard quadratic form \( ax^2 + bx + c = 0 \). The value of the discriminant tells us:- If the discriminant is positive, there are two distinct real solutions.- If the discriminant equals zero, there is exactly one real solution, often called a repeated or double root.- If the discriminant is negative, there are no real solutions, but two complex solutions.For example, in the quadratic equation \( 3x^2 - 15x + 18 = 0 \), the discriminant is calculated as \( 9 \). Since it's positive, the equation has two distinct real solutions. Always compute the discriminant first to guide you in selecting the appropriate method for solving the quadratic equation.

Factoring

Factoring is another method to solve quadratic equations, where the equation is rewritten as a product of simpler expressions set equal to zero. The standard approach is to find two numbers that multiply to the constant term \( c \) and add to the linear coefficient \( b \). For example, solving \( 3x^2 - 15x + 18 = 0 \) can also begin by checking if it factors easily.The steps to factor include:- Write the equation in standard form \( ax^2 + bx + c = 0 \).- Find pairs that multiply to \( ac \) (\( 3 \times 18 = 54 \)) and sum to \( -15 \).- Use these pairs to rewrite the middle term and group terms for factoring.If factoring isn't feasible, as is often the case with more complex quadratics, using the quadratic formula is the next step.

Standard Form of Quadratic Equation

The standard form of a quadratic equation is expressed as \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). This form is crucial because it allows for the consistent application of mathematical procedures like factoring, calculating the discriminant, and using the quadratic formula.To convert a quadratic equation to standard form, ensure all terms are on one side of the equation and set to zero. For example, the equation \( 3x^2 + 18 = 15x \) is converted by subtracting \( 15x \) from both sides, yielding \( 3x^2 - 15x + 18 = 0 \).Maintaining the equation in this form simplifies identifying the coefficients needed for further solving methods. It sets the stage for exploring the nature of the quadratic through its discriminant and solutions.