To fill in the ??? in the expression for the quadratic formula, we'll calculate the value of the discriminant, which is given by \( b^2 - 4ac \).
In your equation \( 2x^2 + 7x - 15 = 0 \):
- \( a = 2 \)
- \( b = 7 \)
- \( c = -15 \)
Now, plug these values into the discriminant formula:
\[ b^2 - 4ac = 7^2 - 4(2)(-15) \]
Calculating \( 7^2 \):
\[ 7^2 = 49 \]
Now calculating \( -4(2)(-15) \):
\[ -4(2)(-15) = 8 \times 15 = 120 \]
Now, combine these results:
\[ 49 + 120 = 169 \]
So, the expression for the discriminant is \( 169 \). Therefore, the complete equation with the discriminant filled in is:
\[ x = \frac{-7 \pm \sqrt{169}}{4} \]
Thus, the number that goes in place of ??? is \( 169 \).