Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x+12/6=14
-
Equation |x-4|=2
-
Equation -8x^2=0
-
Equation t^2+t-1=0
- Express {x} in terms of y where:
- -18*x-19*y=-16
- 8*x+2*y=-6
- -3*x-3*y=14
- -9*x+17*y=-1
-
Identical expressions
- t^ two +t- one = zero
- t squared plus t minus 1 equally 0
- t to the power of two plus t minus one equally zero
- t2+t-1=0
- t²+t-1=0
- t to the power of 2+t-1=0
- t^2+t-1=O
-
Similar expressions
- t^2-t-1=0
- t^2+t+1=0
t^2+t-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:
$$t_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$t_{2} = \frac{- \sqrt{D} - b}{2 a}$$
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
$$t_{1} = - \frac{1}{2} + \frac{\sqrt{5}}{2}$$
$$t_{2} = - \frac{\sqrt{5}}{2} - \frac{1}{2}$$
a*t^2 + b*t + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$t_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$t_{2} = \frac{- \sqrt{D} - b}{2 a}$$
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.
t1 = (-b + sqrt(D)) / (2*a)
t2 = (-b - sqrt(D)) / (2*a)
or
$$t_{1} = - \frac{1}{2} + \frac{\sqrt{5}}{2}$$
$$t_{2} = - \frac{\sqrt{5}}{2} - \frac{1}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p t + q + t^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = -1$$
Vieta Formulas
$$t_{1} + t_{2} = - p$$
$$t_{1} t_{2} = q$$
$$t_{1} + t_{2} = -1$$
$$t_{1} t_{2} = -1$$
$$p t + q + t^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = -1$$
Vieta Formulas
$$t_{1} + t_{2} = - p$$
$$t_{1} t_{2} = q$$
$$t_{1} + t_{2} = -1$$
$$t_{1} t_{2} = -1$$
Rapid solution
[src]
___
1 \/ 5
t1 = - - + -----
2 2
$$t_{1} = - \frac{1}{2} + \frac{\sqrt{5}}{2}$$
___
1 \/ 5
t2 = - - - -----
2 2
$$t_{2} = - \frac{\sqrt{5}}{2} - \frac{1}{2}$$
t2 = -sqrt(5)/2 - 1/2
Sum and product of roots
[src]
sum
___ ___ 1 \/ 5 1 \/ 5 - - + ----- + - - - ----- 2 2 2 2
$$\left(- \frac{\sqrt{5}}{2} - \frac{1}{2}\right) + \left(- \frac{1}{2} + \frac{\sqrt{5}}{2}\right)$$
=
-1
$$-1$$
product
/ ___\ / ___\ | 1 \/ 5 | | 1 \/ 5 | |- - + -----|*|- - - -----| \ 2 2 / \ 2 2 /
$$\left(- \frac{1}{2} + \frac{\sqrt{5}}{2}\right) \left(- \frac{\sqrt{5}}{2} - \frac{1}{2}\right)$$
=
-1
$$-1$$
-1
The graph