Problem 36
Question
\(p^{2}-p-72=0\)
Step-by-Step Solution
Verified Answer
The solutions are: \( p = 9 \) and \( p = -8 \).
1Step 1 - Write down the quadratic equation
The given equation is: \[ p^{2} - p - 72 = 0 \]
2Step 2 - Identify coefficients
Determine the coefficients in the equation. Here, the equation is in the form \( ax^2 + bx + c = 0 \), where: \[ a = 1, \, b = -1, \, c = -72 \]
3Step 3 - Find the roots using the quadratic formula
Use the quadratic formula \( p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Plug in the values of the coefficients: \[ a = 1, \, b = -1, \, c = -72 \] \[ p = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-72)}}{2 \cdot 1} \]
4Step 4 - Simplify inside the square root
Calculate the value inside the square root: \[ (-1)^2 - 4 \cdot 1 \cdot (-72) = 1 + 288 = 289 \]
5Step 5 - Calculate the square root of 289
The square root of 289 is 17. So, we now have: \[ p = \frac{1 \pm 17}{2} \]
6Step 6 - Solve for the values of p
Calculate the two possible values for \( p \): \[ p = \frac{1 + 17}{2} = 9 \] \[ p = \frac{1 - 17}{2} = -8 \]
7Step 7 - Write the solutions
The solutions to the equation \( p^{2} - p - 72 = 0 \) are: \[ p = 9 \] and \[ p = -8 \]
Key Concepts
quadratic formulafinding rootssimplifying expressions
quadratic formula
The quadratic formula is an essential tool for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). This universal formula allows you to find the roots (or solutions) of any quadratic equation. The formula is:
\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here’s a simple breakdown for using the formula:
E.g., for the equation \( p^2 - p - 72 = 0 \), the coefficients are \( a = 1 \), \( b = -1 \), and \( c = -72 \). Plugging these into the quadratic formula gives two possible solutions for \( p \).
\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here’s a simple breakdown for using the formula:
- Identify the coefficients \( a \), \( b \), and \( c \) from your quadratic equation.
- Plug the values of \( a \), \( b \), and \( c \) into the formula.
- Compute the value inside the square root, known as the discriminant: \( b^2 - 4ac \).
- Solve the equation by calculating the two possible values for \( x \), using the plus and minus options in the formula.
E.g., for the equation \( p^2 - p - 72 = 0 \), the coefficients are \( a = 1 \), \( b = -1 \), and \( c = -72 \). Plugging these into the quadratic formula gives two possible solutions for \( p \).
finding roots
Finding the roots of a quadratic equation means identifying the values of the variable that make the equation true. Roots are where the graph of the equation touches the x-axis.
Using the quadratic formula, we substituted the coefficients from the equation \( p^2 - p - 72 = 0 \) into:
\[ p = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-72)}}{2 \cdot 1} \]
Next, simplify inside the square root:
\(((-1)^2 - 4 \cdot 1 \cdot (-72)) = 1 + 288 = 289 \).
Now compute the square root of 289, which is 17.
So,\[ p = \frac{1 \pm 17}{2} \]
This results in two potential solutions or roots:
Thus, the roots of the equation are \( p = 9 \) and \( p = -8 \). These are the values of \( p \) that satisfy the original equation.
Using the quadratic formula, we substituted the coefficients from the equation \( p^2 - p - 72 = 0 \) into:
- \( a = 1 \)
- \( b = -1 \)
- \( c = -72 \)
\[ p = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-72)}}{2 \cdot 1} \]
Next, simplify inside the square root:
\(((-1)^2 - 4 \cdot 1 \cdot (-72)) = 1 + 288 = 289 \).
Now compute the square root of 289, which is 17.
So,\[ p = \frac{1 \pm 17}{2} \]
This results in two potential solutions or roots:
- \( p = \frac{1 + 17}{2} = 9 \)
- \( p = \frac{1 - 17}{2} = -8 \)
Thus, the roots of the equation are \( p = 9 \) and \( p = -8 \). These are the values of \( p \) that satisfy the original equation.
simplifying expressions
Simplifying expressions is a key step in solving quadratic equations. It involves performing arithmetic operations to make the equation easier to solve.
Here's the step-by-step simplification process applied to our equation:
1. **Identify the Discriminant**: The discriminant \( \Delta \) is the part under the square root in the quadratic formula: \( b^2 - 4ac \).
For \( p^2 - p - 72 = 0 \), the discriminant is:
\[ (-1)^2 - 4 \cdot 1 \cdot (-72) = 1 + 288 = 289 \].
2. **Calculate the Square Root**: Next, find the square root of 289, which is 17. This step is crucial as it helps in splitting the equation into two scenarios:
\[ p = \frac{1 \pm 17}{2} \].
3. **Split into Two Equations**: Use the \( \pm \) symbol to split the equation:
Here's the step-by-step simplification process applied to our equation:
1. **Identify the Discriminant**: The discriminant \( \Delta \) is the part under the square root in the quadratic formula: \( b^2 - 4ac \).
For \( p^2 - p - 72 = 0 \), the discriminant is:
\[ (-1)^2 - 4 \cdot 1 \cdot (-72) = 1 + 288 = 289 \].
2. **Calculate the Square Root**: Next, find the square root of 289, which is 17. This step is crucial as it helps in splitting the equation into two scenarios:
\[ p = \frac{1 \pm 17}{2} \].
3. **Split into Two Equations**: Use the \( \pm \) symbol to split the equation:
- \( p = \frac{1 + 17}{2} = 9 \)
- \( p = \frac{1 - 17}{2} = -8 \)