Question

Use the quadratic formula to solve. Express your answer in simplest form.
7, w, squared, plus, 2, w, minus, 3, equals, minus, w, squared
7w
2
+2w−3=
−w
2

Answer 1

To solve the equation \( 7w^2 + 2w - 3 = -w^2 \) using the quadratic formula, we first need to rearrange the equation into standard form \( ax^2 + bx + c = 0 \).

1. Start by adding \( w^2 \) to both sides of the equation:

\[
7w^2 + 2w - 3 + w^2 = 0
\]

This simplifies to:

\[
8w^2 + 2w - 3 = 0
\]

2. Identify the coefficients \( a \), \( b \), and \( c \):

- \( a = 8 \)
- \( b = 2 \)
- \( c = -3 \)

3. Apply the quadratic formula \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

4. Calculate the discriminant \( b^2 - 4ac \):

\[
b^2 - 4ac = (2)^2 - 4(8)(-3) = 4 + 96 = 100
\]

5. Now substitute \( b \) and the discriminant into the quadratic formula:

\[
w = \frac{-2 \pm \sqrt{100}}{2 \cdot 8}
\]

6. Simplify further:

\[
w = \frac{-2 \pm 10}{16}
\]

Now we will find the two possible values for \( w \):

- For the "plus" case:

\[
w = \frac{-2 + 10}{16} = \frac{8}{16} = \frac{1}{2}
\]

- For the "minus" case:

\[
w = \frac{-2 - 10}{16} = \frac{-12}{16} = -\frac{3}{4}
\]

7. So, the solutions to the equation \( 7w^2 + 2w - 3 = -w^2 \) are:

\[
w = \frac{1}{2} \quad \text{and} \quad w = -\frac{3}{4}
\]