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6x^2-6x^1-36x-40=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
2 1 6*x - 6*x - 36*x - 40 = 0
$$\left(- 36 x + \left(6 x^{2} - 6 x^{1}\right)\right) - 40 = 0$$
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
$$a = 6$$
$$b = -42$$
$$c = -40$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{7}{2} + \frac{\sqrt{681}}{6}$$
$$x_{2} = \frac{7}{2} - \frac{\sqrt{681}}{6}$$
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
$$a = 6$$
$$b = -42$$
$$c = -40$$
, then
D = b^2 - 4 * a * c =
(-42)^2 - 4 * (6) * (-40) = 2724
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{7}{2} + \frac{\sqrt{681}}{6}$$
$$x_{2} = \frac{7}{2} - \frac{\sqrt{681}}{6}$$
Vieta's Theorem
rewrite the equation
$$\left(- 36 x + \left(6 x^{2} - 6 x^{1}\right)\right) - 40 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 7 x - \frac{20}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -7$$
$$q = \frac{c}{a}$$
$$q = - \frac{20}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 7$$
$$x_{1} x_{2} = - \frac{20}{3}$$
$$\left(- 36 x + \left(6 x^{2} - 6 x^{1}\right)\right) - 40 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 7 x - \frac{20}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -7$$
$$q = \frac{c}{a}$$
$$q = - \frac{20}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 7$$
$$x_{1} x_{2} = - \frac{20}{3}$$
Sum and product of roots
[src]
sum
_____ _____ 7 \/ 681 7 \/ 681 - - ------- + - + ------- 2 6 2 6
$$\left(\frac{7}{2} - \frac{\sqrt{681}}{6}\right) + \left(\frac{7}{2} + \frac{\sqrt{681}}{6}\right)$$
=
7
$$7$$
product
/ _____\ / _____\ |7 \/ 681 | |7 \/ 681 | |- - -------|*|- + -------| \2 6 / \2 6 /
$$\left(\frac{7}{2} - \frac{\sqrt{681}}{6}\right) \left(\frac{7}{2} + \frac{\sqrt{681}}{6}\right)$$
=
-20/3
$$- \frac{20}{3}$$
-20/3
Rapid solution
[src]
_____
7 \/ 681
x1 = - - -------
2 6
$$x_{1} = \frac{7}{2} - \frac{\sqrt{681}}{6}$$
_____
7 \/ 681
x2 = - + -------
2 6
$$x_{2} = \frac{7}{2} + \frac{\sqrt{681}}{6}$$
x2 = 7/2 + sqrt(681)/6