Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation -x^2-4*x-3=0
-
Equation (x-4)/x=(2*x+10)/(x+4)
-
Equation x^4=(3x-10)^2
-
Equation -14x+2=-4x+32
- Express {x} in terms of y where:
- -1*x+1*y=-13
- -19*x+20*y=-11
- 4*x+9*y=4
- 8*x+2*y=-11
- Factor polynomial:
- x^2-x-1
- Factor squared:
- x^2-x-1
- Graphing y =:
- x^2-x-1
-
Identical expressions
- x^ two -x- one = zero
- x squared minus x minus 1 equally 0
- x to the power of two minus x minus one equally zero
- x2-x-1=0
- x²-x-1=0
- x to the power of 2-x-1=0
- x^2-x-1=O
-
Similar expressions
- x^2-x+1=0
- x^2+x-1=0
x^2-x-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
$$a = 1$$
$$b = -1$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{5}}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{5}}{2}$$
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
$$a = 1$$
$$b = -1$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(-1)^2 - 4 * (1) * (-1) = 5
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{5}}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{5}}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = -1$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = -1$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = -1$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = -1$$
Rapid solution
[src]
___
1 \/ 5
x1 = - - -----
2 2
$$x_{1} = \frac{1}{2} - \frac{\sqrt{5}}{2}$$
___
1 \/ 5
x2 = - + -----
2 2
$$x_{2} = \frac{1}{2} + \frac{\sqrt{5}}{2}$$
x2 = 1/2 + sqrt(5)/2
Sum and product of roots
[src]
sum
___ ___ 1 \/ 5 1 \/ 5 - - ----- + - + ----- 2 2 2 2
$$\left(\frac{1}{2} - \frac{\sqrt{5}}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right)$$
=
1
$$1$$
product
/ ___\ / ___\ |1 \/ 5 | |1 \/ 5 | |- - -----|*|- + -----| \2 2 / \2 2 /
$$\left(\frac{1}{2} - \frac{\sqrt{5}}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right)$$
=
-1
$$-1$$
-1
The graph