how i solve this f(x)=0.3582*x^2 - 3.3833 * x + 9.0748
When asked to solve f(x) = 0, that means find the value(s) of x that, when substituted into the expression for f(x) will make it equal to zero.
So, to find f(x) = 0, then we set 0.3582*x^2 - 3.3833 * x + 9.0748 = 0.
This is a quadratic equation, of the form ax^2 + bx + c = 0, with a = 0.3582, b = - 3.3833 and c = 9.0748.
We use the quadratic formula to solve quadratic equations. This is,
x = {-b +/- sqrt(b^2 - 4ac)} / (2a)
substituting for the values given for a, b and c,
x = {3.3833 +/- sqrt((-3.3833)^2 - 4*(0.3582)*(9.0748))} / (2*0.3582)
x = {3.3833 +/- sqrt(11.4467 - 13.0024)} / (0.7164)
x = {3.3833 +/- sqrt(-1.55565)} / (0.7164)
Since the discriminant is negative (-1.55565), that means that we have no real values for x.
The expression, 0.3582*x^2 - 3.3833 * x + 9.0748, is a polynomial of 2nd degree. If you were to plot this curve you would get a parabola (U-shaped upwards) and its vertex would be above the x-axis. That means the curve never crosses the x-axis, which is why there is no (real) solution.
Our solution(s) then are complex numbers, made up of real and imaginary parts.
We write the square root part as: sqrt(-1.55565) = sqrt(1.55565) * sqrt(-1) = sqrt(1.55565)}*i,
where i = sqrt(-1).
So now we have,
x = {3.3833 +/- sqrt(1.55565)*i} / (0.7164)
x = {3.3833 +/- 1.24726*i} / (0.7164)
x = 4.72264 +/- 1.741*i