Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^2+9=(x+9)^2
-
Equation -x=4-3(3x-8)
-
Equation 9x-7=6x+14
-
Equation x+3=-9*x
- Express {x} in terms of y where:
- -18*x-1*y=-11
- 1*x-6*y=-19
- -1*x-20*y=-17
- 7*x+13*y=-13
- Graphing y =:
- x^2-2x-1
- Integral of d{x}:
-
x^2-2x-1
-
Identical expressions
- x^ two -2x- one = zero
- x squared minus 2x minus 1 equally 0
- x to the power of two minus 2x minus one equally zero
- x2-2x-1=0
- x²-2x-1=0
- x to the power of 2-2x-1=0
- x^2-2x-1=O
-
Similar expressions
- x^2-2x+1=0
- x^2+2x-1=0
- (3x^2-2*x-1)*(3x-1)=0
x^2-2x-1=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
$$a = 1$$
$$b = -2$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 1 + \sqrt{2}$$
$$x_{2} = 1 - \sqrt{2}$$
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
$$a = 1$$
$$b = -2$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(-2)^2 - 4 * (1) * (-1) = 8
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 1 + \sqrt{2}$$
$$x_{2} = 1 - \sqrt{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -2$$
$$q = \frac{c}{a}$$
$$q = -1$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 2$$
$$x_{1} x_{2} = -1$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -2$$
$$q = \frac{c}{a}$$
$$q = -1$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 2$$
$$x_{1} x_{2} = -1$$
Rapid solution
[src]
___ x1 = 1 - \/ 2
$$x_{1} = 1 - \sqrt{2}$$
___ x2 = 1 + \/ 2
$$x_{2} = 1 + \sqrt{2}$$
x2 = 1 + sqrt(2)
Sum and product of roots
[src]
sum
___ ___ 1 - \/ 2 + 1 + \/ 2
$$\left(1 - \sqrt{2}\right) + \left(1 + \sqrt{2}\right)$$
=
2
$$2$$
product
/ ___\ / ___\ \1 - \/ 2 /*\1 + \/ 2 /
$$\left(1 - \sqrt{2}\right) \left(1 + \sqrt{2}\right)$$
=
-1
$$-1$$
-1
The graph