Mister Exam
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How to use it?
- Equation:
- Equation ax=b
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Equation x^2-x+8=0
- Equation x^2+y^2=1
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Equation x^2-8x+16=0
- Express {x} in terms of y where:
- -4*x+7*y=-16
- -1*x-20*y=1
- 14*x-10*y=-4
- -11*x+16*y=3
-
Identical expressions
- x^ two -x+ eight = zero
- x squared minus x plus 8 equally 0
- x to the power of two minus x plus eight equally zero
- x2-x+8=0
- x²-x+8=0
- x to the power of 2-x+8=0
- x^2-x+8=O
-
Similar expressions
- x^2-x-8=0
- x^2+x+8=0
x^2-x+8=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
$$a = 1$$
$$b = -1$$
$$c = 8$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{31} i}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{31} i}{2}$$
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
$$a = 1$$
$$b = -1$$
$$c = 8$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (8) = -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
$$x_{1} = \frac{1}{2} + \frac{\sqrt{31} i}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{31} i}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 8$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = 8$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = 8$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = 8$$
Sum and product of roots
[src]
sum
____ ____ 1 I*\/ 31 1 I*\/ 31 - - -------- + - + -------- 2 2 2 2
$$\left(\frac{1}{2} - \frac{\sqrt{31} i}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{31} i}{2}\right)$$
=
1
$$1$$
product
/ ____\ / ____\ |1 I*\/ 31 | |1 I*\/ 31 | |- - --------|*|- + --------| \2 2 / \2 2 /
$$\left(\frac{1}{2} - \frac{\sqrt{31} i}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{31} i}{2}\right)$$
=
8
$$8$$
8
Rapid solution
[src]
____
1 I*\/ 31
x1 = - - --------
2 2
$$x_{1} = \frac{1}{2} - \frac{\sqrt{31} i}{2}$$
____
1 I*\/ 31
x2 = - + --------
2 2
$$x_{2} = \frac{1}{2} + \frac{\sqrt{31} i}{2}$$
x2 = 1/2 + sqrt(31)*i/2
Numerical answer
[src]
x1 = 0.5 - 2.78388218141501*i
x2 = 0.5 + 2.78388218141501*i
x2 = 0.5 + 2.78388218141501*i
The graph