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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.
What else can I help you with?
Which two values of x are roots of the polynomial x2 plus 5x plus 9?
-2.5 + 1.6583123951777i-2.5 - 1.6583123951777i
Which two values of x are roots of the polynomial below x2 plus 5x plus 11?
There are none because the discriminant of the given quadratic expression is less than zero.
X2-11x plus 13 has two values of x are roots of a polynomial what is it?
x=11+69/2 and x=11-69/2
Which two values of x are roots of the polynomial x2 plus 3x - 5?
To find the roots of the polynomial (x^2 + 3x - 5), we need to set the polynomial equal to zero and solve for x. So, (x^2 + 3x - 5 = 0). To solve this quadratic equation, we can use the quadratic formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}), where a = 1, b = 3, and c = -5. Plugging these values into the formula, we get (x = \frac{-3 \pm \sqrt{3^2 - 41(-5)}}{2*1}), which simplifies to (x = \frac{-3 \pm \sqrt{29}}{2}). Therefore, the two values of x that are roots of the polynomial are (x = \frac{-3 + \sqrt{29}}{2}) and (x = \frac{-3 - \sqrt{29}}{2}).
What kind of polynomial is this 2x5 plus 2x4 plus 2x plus 1?
A fifth degree polynomial.
Which two values of x are roots of the polynomial x2 plus 5x plus 9?
-2.5 + 1.6583123951777i-2.5 - 1.6583123951777i
Which two values of x are roots of the polynomial below x2 plus 5x plus 11?
There are none because the discriminant of the given quadratic expression is less than zero.
What 2 values of x are roots of the polynomial x2 plus 3x-5?
You can find the roots with the quadratic equation (a = 1, b = 3, c = -5).
Which two values of x are roots of the polynomial below x2 plus 5x plus 9?
x = -2.5 + 1.6583123951777ix = -2.5 - 1.6583123951777iwhere i is the square root of negative one.
The polynomial 32 plus 4x plus 3 has how many roots?
1
X2-11x plus 13 has two values of x are roots of a polynomial what is it?
x=11+69/2 and x=11-69/2
If the factors of a polynomial are x plus 2 and x plus 6 what values of x make that polynomial 0?
-2 and -6
What are the roots of the polynomial x2 plus 3x plus 5?
You can find the roots with the quadratic equation (a = 1, b = 3, c = -5).
The polynomial 4x2 plus 5x plus 4 has how many roots?
None, it involves the square root of a negative number so the roots are imaginary.
Which two values of x are roots of the polynomial of x2 plus 5x plus 9?
To find the roots of the polynomial (x^2 + 5x + 9), we can use the quadratic formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}). Here, (a = 1), (b = 5), and (c = 9). The discriminant (b^2 - 4ac = 5^2 - 4 \cdot 1 \cdot 9 = 25 - 36 = -11), which is negative. This means the polynomial has no real roots, but two complex roots: (x = \frac{-5 \pm i\sqrt{11}}{2}).
If the factors of a polynomial are x plus 4 and x plus 8 what values of x make that polynomial 0?
-6 Check: -6+4-6+8 = 0
What is 4x squared plus 16x plus 15?
The expression (4x^2 + 16x + 15) is a quadratic polynomial. It can be factored into ((2x + 3)(2x + 5)). Alternatively, it can be analyzed using the quadratic formula to find its roots or solved for specific values of (x).
