Problem 38 - Applied Calculus

Question

Solve each equation by factoring or the Quadratic Formula, as appropriate. $$ x^{2}-3 x=54 $$

Step-by-Step Solution

Answer

The solutions are \( x = 9 \) and \( x = -6 \).

Step 1: Move all terms to one side

First, rearrange the given equation to have all terms on one side so that one side becomes zero. Therefore, subtract 54 from both sides:\[ x^2 - 3x - 54 = 0 \]

Step 2: Identify if factoring is possible

We will check if we can factor the quadratic equation by looking for two numbers that multiply to -54 and add up to -3. After checking, we find that no integer pairs satisfy this condition, so we will use the Quadratic Formula instead.

Step 3: Apply the Quadratic Formula

The Quadratic Formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For the equation \( x^2 - 3x - 54 = 0 \), we identify \( a = 1 \), \( b = -3 \), and \( c = -54 \). Substitute these values into the formula.

Step 4: Calculate discriminant

Calculate the discriminant \( b^2 - 4ac \):\[ (-3)^2 - 4(1)(-54) = 9 + 216 = 225 \]

Step 5: Solve for x using the Quadratic Formula

Since \( \sqrt{225} = 15 \), substitute back into the formula:\[ x = \frac{-(-3) \pm 15}{2(1)} \]This simplifies to:\[ x = \frac{3 \pm 15}{2} \]

Step 6: Find the solutions

Calculate the two possible values for \( x \):- \( x_1 = \frac{3 + 15}{2} = 9 \)- \( x_2 = \frac{3 - 15}{2} = -6 \)Thus, the solutions are \( x = 9 \) and \( x = -6 \).

Key Concepts

Factoring Quadratic Equations

Factoring quadratic equations is a technique used to find the roots or solutions of quadratic equations. When you have a quadratic equation of the form \( ax^2 + bx + c = 0 \), you try to express it as a product of two binomials.
For example, for the equation \( x^2 - 3x - 54 = 0 \), factoring would mean finding two numbers that multiply to -54 (the constant term) and add to -3 (the coefficient of \( x \)).

• The first step usually involves identifying potential pairs of factors of the constant term, which in this case is -54.
• This can be a trial-and-error process, where you try different pairs to see if they work.
• However, if the numbers don’t quite match, then factoring might not be the best method.

In situations where factoring is not possible, or is difficult, quadratic formulas can be used as an effective alternative to find solutions.

Discriminant Calculation

The discriminant is an essential part of the quadratic formula, giving us vital information about the nature of the roots of the quadratic equation. It is calculated as \( b^2 - 4ac \) from the quadratic equation \( ax^2 + bx + c = 0 \).
For our equation \( x^2 - 3x - 54 = 0 \), we have calculated the discriminant as follows:

• \( a = 1 \), \( b = -3 \), \( c = -54 \)
• So, \( (-3)^2 - 4(1)(-54) = 9 + 216 = 225 \)

The value of the discriminant helps predict the type and number of solutions:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root, a repeated root.
• If the discriminant is negative, there are no real roots, only complex ones.

Here, a positive discriminant of 225 indicates two distinct real solutions.

Solving Quadratic Equations Step by Step

Solving quadratic equations step by step ensures accuracy and enhances understanding of the overall process. Let's break down the process we used with the equation \( x^2 - 3x - 54 = 0 \):

• Step 1: Rearrange. Begin by rearranging all terms to one side of the equation, forming \( x^2 - 3x - 54 = 0 \) with zero on the other side.
• Step 2: Factoring Check. Evaluate whether the equation can be factored. This involves looking for two numbers that are factors of -54 that add up to -3. If this isn't possible, go directly to the quadratic formula.
• Step 3: Apply the Quadratic Formula. Use \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with values \( a = 1 \), \( b = -3 \), \( c = -54 \).
• Step 4: Discriminant Calculation. A discriminant of 225 shows us there are two distinct real roots.
• Step 5: Solve for x. Replace in the formula and simplify to find \( x = 9 \) and \( x = -6 \), the solutions to the equation.

Following these steps, we effectively solved the equation. This systematic method is particularly useful when tackling more complex quadratic equations.