Mister Exam
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Equation 7x^2-2x+12=0
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- 7x^2-2x-12=0
- 7x^2+2x+12=0
7x^2-2x+12=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
$$a = 7$$
$$b = -2$$
$$c = 12$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{1}{7} + \frac{\sqrt{83} i}{7}$$
Simplify
$$x_{2} = \frac{1}{7} - \frac{\sqrt{83} i}{7}$$
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 - it is the discriminant.
Because
$$a = 7$$
$$b = -2$$
$$c = 12$$
, then
D = b^2 - 4 * a * c =
(-2)^2 - 4 * (7) * (12) = -332
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{1}{7} + \frac{\sqrt{83} i}{7}$$
Simplify
$$x_{2} = \frac{1}{7} - \frac{\sqrt{83} i}{7}$$
Simplify
Vieta's Theorem
rewrite the equation
$$7 x^{2} - 2 x + 12 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{2 x}{7} + \frac{12}{7} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{2}{7}$$
$$q = \frac{c}{a}$$
$$q = \frac{12}{7}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{2}{7}$$
$$x_{1} x_{2} = \frac{12}{7}$$
$$7 x^{2} - 2 x + 12 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{2 x}{7} + \frac{12}{7} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{2}{7}$$
$$q = \frac{c}{a}$$
$$q = \frac{12}{7}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{2}{7}$$
$$x_{1} x_{2} = \frac{12}{7}$$
Sum and product of roots
[src]
sum
____ ____
1 I*\/ 83 1 I*\/ 83
0 + - - -------- + - + --------
7 7 7 7
$$\left(0 + \left(\frac{1}{7} - \frac{\sqrt{83} i}{7}\right)\right) + \left(\frac{1}{7} + \frac{\sqrt{83} i}{7}\right)$$
=
2/7
$$\frac{2}{7}$$
product
/ ____\ / ____\ |1 I*\/ 83 | |1 I*\/ 83 | 1*|- - --------|*|- + --------| \7 7 / \7 7 /
$$1 \cdot \left(\frac{1}{7} - \frac{\sqrt{83} i}{7}\right) \left(\frac{1}{7} + \frac{\sqrt{83} i}{7}\right)$$
=
12/7
$$\frac{12}{7}$$
12/7
Rapid solution
[src]
____
1 I*\/ 83
x1 = - - --------
7 7
$$x_{1} = \frac{1}{7} - \frac{\sqrt{83} i}{7}$$
____
1 I*\/ 83
x2 = - + --------
7 7
$$x_{2} = \frac{1}{7} + \frac{\sqrt{83} i}{7}$$
Numerical answer
[src]
x1 = 0.142857142857143 + 1.30149051130633*i
x2 = 0.142857142857143 - 1.30149051130633*i
x2 = 0.142857142857143 - 1.30149051130633*i
The graph