To identify the correct substitution of values into the quadratic formula for the equation \(2p^2 - 7p - 13 = 0\), we first need to identify the coefficients \(a\), \(b\), and \(c\) from the standard form of a quadratic equation \(ap^2 + bp + c = 0\).
From the given equation:
- \(a = 2\)
- \(b = -7\)
- \(c = -13\)
The quadratic formula is given by:
\[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Now, substituting the values of \(a\), \(b\), and \(c\) into the formula:
- For \( -b \): Substitute \(b = -7\). Thus, \(-b = -(-7) = 7\).
- For \(b^2\): Substitute \(b = -7\), so \(b^2 = (-7)^2 = 49\).
- For \(4ac\): Substitute \(a = 2\) and \(c = -13\). Therefore, \(4ac = 4(2)(-13) = -104\).
Putting it together, we have:
\[ p = \frac{7 \pm \sqrt{49 - (-104)}}{2(2)} \]
This means we are looking for a substitution of:
- The coefficient of p (which was fulfilled by \(-(-7)\) or \(7\)).
- The 2 values of \(2a\) which is \(2(2)\).
- The values inside the square root where \((-7)^2 - 4(2)(-13)\).
Now let's analyze each option provided:
-
First option:
\[ p = \frac{2(-7) \pm \sqrt{(-7)^2 - 4(-2)(-13)}}{2} \] Incorrect substitution of \(a\).
-
Second option:
\[ p = \frac{2(2) - 7 \pm \sqrt{(-7)^2 - 4(2)(13)}}{2} \] Incorrect substitution of \(c\).
-
Third option:
\[ p = \frac{2(2) - (-7) \pm \sqrt{(-7)^2 - 4(2)(13)}}{2} \] Incorrect substitution of \(c\).
-
Fourth option:
\[ p = \frac{2(2) - (-7) \pm \sqrt{(-7)^2 - 4(2)(-13)}}{2} \] Correct substitution. Here \(-(-7)\) correctly gives \(7\) and \(4ac\) is properly given with a negative sign for \(c\).
Thus, the correct substitution is:
Fourth option:
\[ p = \frac{2(2) - (-7) \pm \sqrt{(-7)^2 - 4(2)(-13)}}{2} \]