To solve the equation \( 7w^2 + 2w - 3 = -w^2 \) using the quadratic formula, we first need to rearrange the equation into standard form \( ax^2 + bx + c = 0 \).
- Start by adding \( w^2 \) to both sides of the equation:
\[ 7w^2 + 2w - 3 + w^2 = 0 \]
This simplifies to:
\[ 8w^2 + 2w - 3 = 0 \]
- Identify the coefficients \( a \), \( b \), and \( c \):
- \( a = 8 \)
- \( b = 2 \)
- \( c = -3 \)
-
Apply the quadratic formula \( w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
-
Calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = (2)^2 - 4(8)(-3) = 4 + 96 = 100 \]
- Now substitute \( b \) and the discriminant into the quadratic formula:
\[ w = \frac{-2 \pm \sqrt{100}}{2 \cdot 8} \]
- 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:
\[ w = \frac{-2 - 10}{16} = \frac{-12}{16} = -\frac{3}{4} \]
- So, the solutions to the equation \( 7w^2 + 2w - 3 = -w^2 \) are:
\[ w = \frac{1}{2} \quad \text{and} \quad w = -\frac{3}{4} \]