Mister Exam
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- 121p²+14p-2=0
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121p²+14p+2=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:
$$p_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$p_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 121$$
$$b = 14$$
$$c = 2$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$p_{1} = - \frac{7}{121} + \frac{\sqrt{193} i}{121}$$
$$p_{2} = - \frac{7}{121} - \frac{\sqrt{193} i}{121}$$
a*p^2 + b*p + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$p_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$p_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 121$$
$$b = 14$$
$$c = 2$$
, then
D = b^2 - 4 * a * c =
(14)^2 - 4 * (121) * (2) = -772
Because D<0, then the equation
has no real roots,
but complex roots is exists.
p1 = (-b + sqrt(D)) / (2*a)
p2 = (-b - sqrt(D)) / (2*a)
or
$$p_{1} = - \frac{7}{121} + \frac{\sqrt{193} i}{121}$$
$$p_{2} = - \frac{7}{121} - \frac{\sqrt{193} i}{121}$$
Vieta's Theorem
rewrite the equation
$$\left(121 p^{2} + 14 p\right) + 2 = 0$$
of
$$a p^{2} + b p + c = 0$$
as reduced quadratic equation
$$p^{2} + \frac{b p}{a} + \frac{c}{a} = 0$$
$$p^{2} + \frac{14 p}{121} + \frac{2}{121} = 0$$
$$2 p^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{14}{121}$$
$$q = \frac{c}{a}$$
$$q = \frac{2}{121}$$
Vieta Formulas
$$p_{1} + p_{2} = - p$$
$$p_{1} p_{2} = q$$
$$p_{1} + p_{2} = - \frac{14}{121}$$
$$p_{1} p_{2} = \frac{2}{121}$$
$$\left(121 p^{2} + 14 p\right) + 2 = 0$$
of
$$a p^{2} + b p + c = 0$$
as reduced quadratic equation
$$p^{2} + \frac{b p}{a} + \frac{c}{a} = 0$$
$$p^{2} + \frac{14 p}{121} + \frac{2}{121} = 0$$
$$2 p^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{14}{121}$$
$$q = \frac{c}{a}$$
$$q = \frac{2}{121}$$
Vieta Formulas
$$p_{1} + p_{2} = - p$$
$$p_{1} p_{2} = q$$
$$p_{1} + p_{2} = - \frac{14}{121}$$
$$p_{1} p_{2} = \frac{2}{121}$$
Rapid solution
[src]
_____
7 I*\/ 193
p1 = - --- - ---------
121 121
$$p_{1} = - \frac{7}{121} - \frac{\sqrt{193} i}{121}$$
_____
7 I*\/ 193
p2 = - --- + ---------
121 121
$$p_{2} = - \frac{7}{121} + \frac{\sqrt{193} i}{121}$$
p2 = -7/121 + sqrt(193)*i/121
Sum and product of roots
[src]
sum
_____ _____ 7 I*\/ 193 7 I*\/ 193 - --- - --------- + - --- + --------- 121 121 121 121
$$\left(- \frac{7}{121} - \frac{\sqrt{193} i}{121}\right) + \left(- \frac{7}{121} + \frac{\sqrt{193} i}{121}\right)$$
=
-14 ---- 121
$$- \frac{14}{121}$$
product
/ _____\ / _____\ | 7 I*\/ 193 | | 7 I*\/ 193 | |- --- - ---------|*|- --- + ---------| \ 121 121 / \ 121 121 /
$$\left(- \frac{7}{121} - \frac{\sqrt{193} i}{121}\right) \left(- \frac{7}{121} + \frac{\sqrt{193} i}{121}\right)$$
=
2/121
$$\frac{2}{121}$$
2/121
Numerical answer
[src]
p1 = -0.0578512396694215 + 0.114813586689668*i
p2 = -0.0578512396694215 - 0.114813586689668*i
p2 = -0.0578512396694215 - 0.114813586689668*i