To solve the quadratic equation [tex]\(5w^2 + 3w = 2\)[/tex], follow these steps:
1. Move all terms to one side to set the equation to 0 form:
[tex]\[
5w^2 + 3w - 2 = 0
\][/tex]
2. Identify the coefficients for the quadratic formula:
- [tex]\(a = 5\)[/tex]
- [tex]\(b = 3\)[/tex]
- [tex]\(c = -2\)[/tex]
3. Use the quadratic formula, which is given by:
[tex]\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
4. Calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
Plug in the values:
[tex]\[
\text{Discriminant} = 3^2 - 4(5)(-2) = 9 + 40 = 49
\][/tex]
5. Find the square root of the discriminant:
[tex]\[
\sqrt{49} = 7
\][/tex]
6. Substitute these values into the quadratic formula to find the two possible solutions for [tex]\(w\)[/tex]:
[tex]\[
w = \frac{-3 \pm 7}{2(5)}
\][/tex]
7. Calculate the two solutions:
- For the positive square root:
[tex]\[
w = \frac{-3 + 7}{10} = \frac{4}{10} = 0.4
\][/tex]
- For the negative square root:
[tex]\[
w = \frac{-3 - 7}{10} = \frac{-10}{10} = -1.0
\][/tex]
Therefore, the solutions for the equation [tex]\(5w^2 + 3w - 2 = 0\)[/tex] are:
[tex]\[
w = 0.4 \quad \text{and} \quad w = -1.0
\][/tex]
