Problem 55 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation using any convenient method. \(12 x=x^{2}+27\)

Step-by-Step Solution

Answer

The solutions to the quadratic equation \(12x = x^{2} + 27\) are \(x = 9\) and \(x = 3\).

Step 1: Arrange to Standard Form

We commence by arranging the equation into standard quadratic form. This can be done by moving all terms to one side of the equation: \(x^2 - 12x + 27 = 0\).

Step 2: Apply Quadratic Formula

We can then apply the quadratic formula \(x = \frac{-b ± \sqrt{b^{2}-4ac}}{2a}\), where in this case, \(a=1\), \(b=-12\), and \(c=27\). Hence, the solutions for \(x\) are as follows: \( x = \frac{-(-12) ± \sqrt{(-12)^{2}-4*1*27}}{2*1}.\)

Step 3: Simplifying Results

Once we compute what's under the square root (\( √{144 - 108} \)), we can further simplify our solution for \(x\). This simplification yields: \(x = \frac{12 ± \sqrt{36}}{2}= \frac{12 ± 6}{2}\).

Step 4: Solve for x

Rounding off, we can promptly find the two solutions when we add and subtract: \(x = \frac{12 + 6}{2}=9 \) and \(x = \frac{12 - 6}{2}=3\). The solutions are \(x = 9\) and \(x = 3\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form \( ax^2 + bx + c = 0 \). A quadratic equation is characterized by its highest degree of two, meaning the variable \( x \) is squared. The quadratic formula provides an explicit solution for \( x \), whether the equation can be factored easily or not. The formula is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a \), \( b \), and \( c \) are constants from the equation, and the symbol \( \pm \) indicates that there could be two possible solutions. The term under the square root, \( b^2 - 4ac \), is called the discriminant and determines the nature of the roots:

• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there is exactly one real solution.
• If it is negative, there are no real solutions, but two complex ones.

This formula covers all possible scenarios for solving quadratic equations.

Standard Form

Before applying the quadratic formula, it's important to arrange the quadratic equation into its standard form, \( ax^2 + bx + c = 0 \). Having the equation in this form ensures that all coefficients (\( a \), \( b \), and \( c \)) are clearly identified, which is essential for the quadratic formula to be used accurately. To convert an equation to the standard form:

• Bring all terms to one side of the equation, setting it equal to zero.
• Order the terms by degree, starting with \( x^2 \), then \( x \), and finally the constant term.

For example, transforming \( 12x = x^2 + 27 \) into the standard form involves:

• Subtracting \( 12x \) from both sides to get \( x^2 - 12x + 27 = 0 \).

Once in this form, you can clearly see the values of \( a \), \( b \), and \( c \), which are crucial for applying the quadratic formula.

Solving Equations

Solving quadratic equations effectively involves understanding various methods, but the quadratic formula is one of the most versatile. Here's how to apply it using the equation \( x^2 - 12x + 27 = 0 \):

• Identify the coefficients: \( a = 1 \), \( b = -12 \), \( c = 27 \).
• Substitute these values into the quadratic formula: \( x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \times 1 \times 27}}{2 \times 1} \).
• Calculate the discriminant: \( (-12)^2 - 4 \times 1 \times 27 = 144 - 108 = 36 \).
• Find the square root of the discriminant: \( \sqrt{36} = 6 \).
• Solve the equation for \( x \):
  • \( x = \frac{12 + 6}{2} = 9 \)
  • \( x = \frac{12 - 6}{2} = 3 \)

This approach provides a systematic way to handle any quadratic equation, yielding accurate and complete solution sets.