Mister Exam
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Identical expressions
- x^ two -17x+ eighty = zero
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Similar expressions
- x^2-17x-80=0
- x^2+17x+80=0
x^2-17x+80=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
$$a = 1$$
$$b = -17$$
$$c = 80$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{17}{2} + \frac{\sqrt{31} i}{2}$$
$$x_{2} = \frac{17}{2} - \frac{\sqrt{31} i}{2}$$
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
$$a = 1$$
$$b = -17$$
$$c = 80$$
, then
D = b^2 - 4 * a * c =
(-17)^2 - 4 * (1) * (80) = -31
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
$$x_{1} = \frac{17}{2} + \frac{\sqrt{31} i}{2}$$
$$x_{2} = \frac{17}{2} - \frac{\sqrt{31} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -17$$
$$q = \frac{c}{a}$$
$$q = 80$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 17$$
$$x_{1} x_{2} = 80$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -17$$
$$q = \frac{c}{a}$$
$$q = 80$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 17$$
$$x_{1} x_{2} = 80$$
Sum and product of roots
[src]
sum
____ ____ 17 I*\/ 31 17 I*\/ 31 -- - -------- + -- + -------- 2 2 2 2
$$\left(\frac{17}{2} - \frac{\sqrt{31} i}{2}\right) + \left(\frac{17}{2} + \frac{\sqrt{31} i}{2}\right)$$
=
17
$$17$$
product
/ ____\ / ____\ |17 I*\/ 31 | |17 I*\/ 31 | |-- - --------|*|-- + --------| \2 2 / \2 2 /
$$\left(\frac{17}{2} - \frac{\sqrt{31} i}{2}\right) \left(\frac{17}{2} + \frac{\sqrt{31} i}{2}\right)$$
=
80
$$80$$
80
Rapid solution
[src]
____
17 I*\/ 31
x1 = -- - --------
2 2
$$x_{1} = \frac{17}{2} - \frac{\sqrt{31} i}{2}$$
____
17 I*\/ 31
x2 = -- + --------
2 2
$$x_{2} = \frac{17}{2} + \frac{\sqrt{31} i}{2}$$
x2 = 17/2 + sqrt(31)*i/2
Numerical answer
[src]
x1 = 8.5 - 2.78388218141501*i
x2 = 8.5 + 2.78388218141501*i
x2 = 8.5 + 2.78388218141501*i