Find the roots of the equation 2x^2 - 3x + 1 = 0 using the quadratic formula.
Answers:
To find the roots of the quadratic equation ( 2x^2 - 3x + 1 = 0 ) using the quadratic formula, we can use the formula:
[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
where ( a ), ( b ), and ( c ) are the coefficients from the equation ( ax^2 + bx + c = 0 ).
In this case, we identify:
- ( a = 2 )
- ( b = -3 )
- ( c = 1 )
Now we will substitute these values into the formula.
First, we calculate the discriminant ( b^2 - 4ac ):
[ b^2 - 4ac = (-3)^2 - 4 \cdot 2 \cdot 1 = 9 - 8 = 1 ]
Next, we can substitute ( b ) and the discriminant back into the quadratic formula:
[ x = \frac{-(-3) \pm \sqrt{1}}{2 \cdot 2} ]
This simplifies to:
[ x = \frac{3 \pm 1}{4} ]
Now we can find the two possible values for ( x ):
- For the positive case ( (3 + 1) ):
[ x = \frac{4}{4} = 1 ]
- For the negative case ( (3 - 1) ):
[ x = \frac{2}{4} = \frac{1}{2} ]
Thus, the roots of the equation ( 2x^2 - 3x + 1 = 0 ) are:
[ \boxed{1} \quad \text{and} \quad \boxed{\frac{1}{2}} ]