Mister Exam
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x^2-5x+9=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
This equation is of the form
$$a*x^2 + b*x + c = 0$$
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 1$$
$$b = -5$$
$$c = 9$$
, then
$$D = b^2 - 4 * a * c = $$
$$\left(-1\right) 1 \cdot 4 \cdot 9 + \left(-5\right)^{2} = -11$$
Because D<0, then the equation
has no real roots,
but complex roots is exists.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = \frac{5}{2} + \frac{\sqrt{11} i}{2}$$
Simplify
$$x_{2} = \frac{5}{2} - \frac{\sqrt{11} i}{2}$$
Simplify
$$a*x^2 + b*x + c = 0$$
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 1$$
$$b = -5$$
$$c = 9$$
, then
$$D = b^2 - 4 * a * c = $$
$$\left(-1\right) 1 \cdot 4 \cdot 9 + \left(-5\right)^{2} = -11$$
Because D<0, then the equation
has no real roots,
but complex roots is exists.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = \frac{5}{2} + \frac{\sqrt{11} i}{2}$$
Simplify
$$x_{2} = \frac{5}{2} - \frac{\sqrt{11} i}{2}$$
Simplify
Vieta's Theorem
it is reduced quadratic equation
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = 9$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = 9$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = 9$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = 9$$
Rapid solution
[src]
____
5 I*\/ 11
x_1 = - - --------
2 2
$$x_{1} = \frac{5}{2} - \frac{\sqrt{11} i}{2}$$
____
5 I*\/ 11
x_2 = - + --------
2 2
$$x_{2} = \frac{5}{2} + \frac{\sqrt{11} i}{2}$$
Sum and product of roots
[src]
sum
____ ____ 5 I*\/ 11 5 I*\/ 11 - - -------- + - + -------- 2 2 2 2
$$\left(\frac{5}{2} - \frac{\sqrt{11} i}{2}\right) + \left(\frac{5}{2} + \frac{\sqrt{11} i}{2}\right)$$
=
5
$$5$$
product
____ ____ 5 I*\/ 11 5 I*\/ 11 - - -------- * - + -------- 2 2 2 2
$$\left(\frac{5}{2} - \frac{\sqrt{11} i}{2}\right) * \left(\frac{5}{2} + \frac{\sqrt{11} i}{2}\right)$$
=
9
$$9$$
Numerical answer
[src]
x1 = 2.5 + 1.6583123951777*i
x2 = 2.5 - 1.6583123951777*i
x2 = 2.5 - 1.6583123951777*i
The graph