let f(x) = 3x^2 - 3x + 1. The roots of f is the same as asking for x where f(x) = 0.
So we do it f(x) = 0, 3x^2 - 3x + 1 = 0. Using "complete the square" method
3(x^2 - x) + 1 = 0
3(x^2 + (2 . - 1/2 . x) + (-1/2)^2 -(-1/2)^2) + 1 = 0
3(x - 1/2)^2 -3/4 + 1 = 0
3(x - 1/2)^2 + 1/4 = 0
By the quadratic formula, the solution comes out to be
x = [3+√(-3)]/6 and x = [3-√(-3)]/6
or
x = (3+i√3)/6 and x = (3-i√3)/6
in other words, the solutions are complex numbers.
-2.5 + 1.6583123951777i-2.5 - 1.6583123951777i
There are none because the discriminant of the given quadratic expression is less than zero.
x=11+69/2 and x=11-69/2
x2 + 3x - 5 is an expression, not an equation. An equation may have roots, an expression does not. However, x2 + 3x - 5 = 0 is an equation and its roots are -4.1926 and 1.1926 (approx).
A fifth degree polynomial.
-2.5 + 1.6583123951777i-2.5 - 1.6583123951777i
There are none because the discriminant of the given quadratic expression is less than zero.
You can find the roots with the quadratic equation (a = 1, b = 3, c = -5).
x = -2.5 + 1.6583123951777ix = -2.5 - 1.6583123951777iwhere i is the square root of negative one.
x=11+69/2 and x=11-69/2
1
-2 and -6
x2 + 3x - 5 is an expression, not an equation. An equation may have roots, an expression does not. However, x2 + 3x - 5 = 0 is an equation and its roots are -4.1926 and 1.1926 (approx).
You can find the roots with the quadratic equation (a = 1, b = 3, c = -5).
None, it involves the square root of a negative number so the roots are imaginary.
-6 Check: -6+4-6+8 = 0
It is: 16-20 = -4 which means that the given quadratic expression has no real roots.