To determine the number of sales Martin needs to make to earn [tex]$1,000 given the function \( P(x) = x^2 + 20x - 196 \), we need to solve the equation for \( x \) when \( P(x) = 1000 \).
First, set up the equation:
\[ x^2 + 20x - 196 = 1000 \]
Then, move all terms to one side of the equation to set it to zero:
\[ x^2 + 20x - 196 - 1000 = 0 \]
Simplify:
\[ x^2 + 20x - 1196 = 0 \]
This is a quadratic equation in the standard form \( ax^2 + bx + c = 0 \) where \( a = 1 \), \( b = 20 \), and \( c = -1196 \).
Now, solve this quadratic equation. One method to solve quadratic equations is to use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
1. Calculate the discriminant:
\[ \Delta = b^2 - 4ac \]
\[ \Delta = 20^2 - 4 \cdot 1 \cdot (-1196) \]
\[ \Delta = 400 + 4784 \]
\[ \Delta = 5184 \]
2. Find the roots using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \]
\[ x = \frac{-20 \pm \sqrt{5184}}{2 \cdot 1} \]
\[ x = \frac{-20 \pm 72}{2} \]
3. Simplify each solution:
\[ x_1 = \frac{-20 + 72}{2} \]
\[ x_1 = \frac{52}{2} \]
\[ x_1 = 26 \]
\[ x_2 = \frac{-20 - 72}{2} \]
\[ x_2 = \frac{-92}{2} \]
\[ x_2 = -46 \]
Since \( x \) represents the number of sales, it must be a real, non-negative number. Therefore, we discard the negative solution \( x = -46 \).
Thus, Martin needs to make \(\boxed{26}\) sales to earn $[/tex]1,000.
The result is that Martin needs to make 26 sales to earn $1,000.
