To solve the equation ( 5y^2 - 8y + 2 = -1 ), we first rearrange it to set it equal to zero:
[
5y^2 - 8y + 2 + 1 = 0
]
[
5y^2 - 8y + 3 = 0
]
Next, we can use the quadratic formula to find the values of ( y ). The quadratic formula is given by:
[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
]
In our case, ( a = 5 ), ( b = -8 ), and ( c = 3 ). We first calculate the discriminant ( b^2 - 4ac ):
[
b^2 - 4ac = (-8)^2 - 4(5)(3) = 64 - 60 = 4
]
Now we'll substitute these values into the quadratic formula:
[
y = \frac{-(-8) \pm \sqrt{4}}{2 \cdot 5}
]
[
y = \frac{8 \pm 2}{10}
]
This gives us two possible solutions:
- ( y = \frac{8 + 2}{10} = \frac{10}{10} = 1 )
- ( y = \frac{8 - 2}{10} = \frac{6}{10} = \frac{3}{5} )
Thus, the solutions to the equation ( 5y^2 - 8y + 2 = -1 ) are:
[
y = 1 \quad \text{and} \quad y = \frac{3}{5}
]
