Mister Exam
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0,5x^2-3x-6=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
This equation is of the form
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 - it is the discriminant.
Because
$$a = \frac{1}{2}$$
$$b = -3$$
$$c = -6$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 3 + \sqrt{21}$$
$$x_{2} = 3 - \sqrt{21}$$
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 - it is the discriminant.
Because
$$a = \frac{1}{2}$$
$$b = -3$$
$$c = -6$$
, then
D = b^2 - 4 * a * c =
(-3)^2 - 4 * (1/2) * (-6) = 21
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 3 + \sqrt{21}$$
$$x_{2} = 3 - \sqrt{21}$$
Vieta's Theorem
rewrite the equation
$$\left(\frac{x^{2}}{2} - 3 x\right) - 6 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 6 x - 12 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -6$$
$$q = \frac{c}{a}$$
$$q = -12$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 6$$
$$x_{1} x_{2} = -12$$
$$\left(\frac{x^{2}}{2} - 3 x\right) - 6 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 6 x - 12 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -6$$
$$q = \frac{c}{a}$$
$$q = -12$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 6$$
$$x_{1} x_{2} = -12$$
Rapid solution
[src]
____ x1 = 3 - \/ 21
$$x_{1} = 3 - \sqrt{21}$$
____ x2 = 3 + \/ 21
$$x_{2} = 3 + \sqrt{21}$$
x2 = 3 + sqrt(21)
Sum and product of roots
[src]
sum
____ ____ 3 - \/ 21 + 3 + \/ 21
$$\left(3 - \sqrt{21}\right) + \left(3 + \sqrt{21}\right)$$
=
6
$$6$$
product
/ ____\ / ____\ \3 - \/ 21 /*\3 + \/ 21 /
$$\left(3 - \sqrt{21}\right) \left(3 + \sqrt{21}\right)$$
=
-12
$$-12$$
-12