Problem 32 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation by using the quadratic formula. $$ 6 p-6=p^{2} $$

Step-by-Step Solution

Answer

The roots of the given quadratic equation \(\pmb{6p - 6 = p^2}\) are \(\pmb{p = 3 + \frac{\sqrt{12}}{2}}\) and \(\pmb{p = 3 - \frac{\sqrt{12}}{2}}\).

Step 1: Rewrite the given equation in standard form

First, rewrite the given equation in the standard form of a quadratic equation, i.e., \(ax^2 + bx + c = 0\). The given equation is: \(6p - 6 = p^2\) Subtract \(p^2\) from both sides to obtain the equation in standard form: \(p^2 - 6p + 6 = 0\)

Step 2: Identify the coefficients a, b, and c

Now, we can see that the quadratic equation is in the form \(ax^2 + bx + c = 0\). Identify the coefficients a, b, and c: \(a = 1, b = -6, c = 6\)

Step 3: Apply the quadratic formula

Apply the quadratic formula to find the roots of the equation: \(p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) Plug in the values of a, b, and c in the formula: \(p = \frac{-( - 6) \pm \sqrt{(- 6)^2 - 4(1)(6)}}{2(1)}\)

Step 4: Simplify the expression

Simplify the expression inside the square root and the fraction: \(p = \frac{6 \pm \sqrt{36 - 24}}{2}\) \(p = \frac{6 \pm \sqrt{12}}{2}\)

Step 5: Simplify the roots (if possible)

Finally, if possible, simplify the roots by factoring out any common factors in the numerator and the denominator: \(p = 3\pm\frac{\sqrt{12}}{2}\) So the roots of the given quadratic equation are \(p = 3 + \frac{\sqrt{12}}{2}\) and \(p = 3 - \frac{\sqrt{12}}{2}\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the second degree. It takes the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable. This kind of equation appears frequently in math, and solving it is a key skill. By comparing a real-world problem to this format, you can address various practical issues.
When dealing with a quadratic equation, the highest exponent of the variable is always 2. This tells us it's a quadratic function. Here's why the quadratic equation is essential:

• Many natural phenomena and engineering problems can be modeled using quadratic equations.
• They allow us to predict values and optimize solutions.
• Learning to solve quadratic equations builds problem-solving skills.

Using the quadratic formula, factoring, and completing the square are some methods to solve these equations. In our given problem: \(6p - 6 = p^2\), we transformed it to the standard form \(p^2 - 6p + 6 = 0\). This step is crucial to apply solving techniques effectively.

Coefficients

In a quadratic equation \(ax^2 + bx + c = 0\), the values of \(a\), \(b\), and \(c\) are called coefficients. They play a significant role in determining the properties and the solution of the equation.
The coefficient \(a\) is attached to \(x^2\) and should never be zero, as it makes the equation non-quadratic. Let's understand each coefficient with our equation example:

• \(a = 1\), which indicates the equation has a positive leading term.
• \(b = -6\), the coefficient of \(x\), affects the symmetry and the roots (or zeros) of the equation.
• \(c = 6\), the constant term, can move the graph of the equation up or down, impacting the solution set's values.

The coefficients directly influence the application of the quadratic formula: \(p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Each coefficient is plugged into the formula to calculate the roots. Understanding how coefficients shape the equation is vital for anticipating the graph's behavior and finding solutions.

Simplifying Expressions

Simplifying expressions is a fundamental skill when working with equations. It eases the calculation process and helps achieve cleaner and more understandable results.
In our quadratic equation, after setting it in the standard form \(p^2 - 6p + 6 = 0\), we applied the quadratic formula, resulting in an expression for \(p\). Each part of this expression needed simplification to reach the final solution.
The steps included:

• Calculating the discriminant inside the square root: \(b^2 - 4ac\).
• Substituting \(b = -6\), \(a = 1\), and \(c = 6\) gives \(36 - 24\), simplifying to \(12\).
• Expressing \(\sqrt{12}\) by factoring: this yields \(3\pm\frac{\sqrt{12}}{2}\).

Simplifying helps us to see that each step builds upon the previous one, ensuring an accurate and efficient process to find solutions. It's about combining like terms, reducing fractions, and finding roots to express values in their simplest forms.