Problem 121 - Elementary Algebra

Question

Solve by using the Quadratic Formula. \(p^{2}-6 p-27=0\)

Step-by-Step Solution

Answer

The solutions are \( x = 9 \) and \( x = -3 \).

Step 1 - Identify coefficients

The quadratic equation is given in the form: \(ax^{2} + bx + c = 0\). Here, identify the coefficients: \(a = 1\), \(b = -6\), and \(c = -27\).

Step 2 - Write down the Quadratic Formula

The Quadratic Formula is: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\].

Step 3 - Substitute coefficients into the formula

Substitute 1 for \(a\), -6 for \(b\), and -27 for \(c\) into the quadratic formula: \[ x = \frac{-(-6) \pm \sqrt{(-6)^{2} - 4(1)(-27)}}{2(1)} \].

Step 4 - Simplify the equation under the square root

Calculate the discriminant: \[ (-6)^{2} - 4(1)(-27) = 36 + 108 = 144 \].

Step 5 - Take the square root

The square root of 144 is 12. Thus, the formula now looks like: \[ x = \frac{6 \pm 12}{2} \].

Step 6 - Calculate the two possible solutions

Calculate the solutions: \( x_{1} = \frac{6 + 12}{2} = 9 \) and \( x_{2} = \frac{6 - 12}{2} = -3 \).

Key Concepts

coefficients of a quadratic equation

Before solving a quadratic equation using the Quadratic Formula, it is essential to identify the coefficients of the equation. A quadratic equation is typically written as:
\( ax^2 + bx + c = 0 \).
Here, the coefficients are:

• \(a\): the coefficient of \(x^2\)
• \(b\): the coefficient of \(x\)
• \(c\): the constant term

In the provided equation, \(p^2 - 6p - 27 = 0\), we can see that:

• \(a = 1\) (as there is no number before \(p^2\), it is implied to be 1)
• \(b = -6\)
• \(c = -27\)

Identifying these coefficients correctly is the first step in using the Quadratic Formula to find the roots of the equation.

discriminant

The discriminant is a crucial part of the Quadratic Formula, which can give us information about the nature of the roots of the equation. It is represented by the expression under the square root in the Quadratic Formula:\(b^2 - 4ac \).

• When the discriminant is positive, there are two distinct real roots.
• When the discriminant is zero, there is exactly one real root (a repeated root).
• When the discriminant is negative, there are no real roots, but two complex roots.

In our example, the discriminant is calculated as follows:\( (-6)^2 - 4(1)(-27) = 36 + 108 = 144 \).
Since the discriminant (144) is positive, it indicates that there are two distinct real roots for the equation. Understanding the discriminant helps you predict the type of solutions you will get from the equation.

solving quadratic equations

Solving quadratic equations using the Quadratic Formula is a systematic process. Let's go through each step using our example equation \(p^2 - 6p - 27 = 0\): \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Substitute the identified coefficients: \(a = 1\), \(b = -6\), and \(c = -27\).
\[ x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-27)}}{2(1)} \].
Simplify under the square root: \( (-6)^2 - 4(1)(-27) = 36 + 108 = 144 \).
Calculate the square root: \( \sqrt{144} = 12 \).
Now our equation is: \[ x = \frac{6 \pm 12}{2} \] which gives us two possible solutions:

• \( x_1 = \frac{6 + 12}{2} = 9 \)
• \( x_2 = \frac{6 - 12}{2} = -3 \)

Each step builds on the last, ensuring you simplify correctly at every stage. Practicing this method will make you more comfortable with solving quadratic equations effortlessly.