Question

Suppose you are a solving a quadratic equation using the quadratic formula and this is your work so far:

2x^{2}+7x-15=0

x=\frac{-7\ \pm\sqrt{\left(7\right)^{2}-4\left(2\right)\left(-15\right)}}{2\left(2\right)}

x=\frac{-7\ \pm\sqrt{???}}{4}

what goes in the ???

Answer 1

To find the expression that goes in the ???, we need to calculate the discriminant \( D \) of the quadratic equation, which is given by the formula:

\[
D = b^2 - 4ac
\]

For the equation \( 2x^{2} + 7x - 15 = 0 \), we have:

- \( a = 2 \)
- \( b = 7 \)
- \( c = -15 \)

Now, substituting these values into the discriminant formula:

\[
D = (7)^{2} - 4(2)(-15)
\]

Calculating \( (7)^2 \):

\[
(7)^2 = 49
\]

Now calculating \( 4(2)(-15) \):

\[
4(2)(-15) = -120
\]

So now substitute these values back into the discriminant:

\[
D = 49 - (-120) = 49 + 120 = 169
\]

Now, we can return to the quadratic formula:

\[
x = \frac{-7 \pm \sqrt{???}}{4}
\]

Since we found \( D = 169 \), we can substitute this value into the formula. Thus,

The expression that goes in the ??? is:

\[
169
\]

So, the final output of your work will be:

\[
x = \frac{-7 \pm \sqrt{169}}{4}
\]