Mister Exam
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Similar expressions
- x^2+5*x+1=0
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x^2-5*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
$$a = 1$$
$$b = -5$$
$$c = 1$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{\sqrt{21}}{2} + \frac{5}{2}$$
$$x_{2} = \frac{5}{2} - \frac{\sqrt{21}}{2}$$
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
$$a = 1$$
$$b = -5$$
$$c = 1$$
, then
D = b^2 - 4 * a * c =
(-5)^2 - 4 * (1) * (1) = 21
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{21}}{2} + \frac{5}{2}$$
$$x_{2} = \frac{5}{2} - \frac{\sqrt{21}}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = 1$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = 1$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -5$$
$$q = \frac{c}{a}$$
$$q = 1$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 5$$
$$x_{1} x_{2} = 1$$
Sum and product of roots
[src]
sum
____ ____ 5 \/ 21 5 \/ 21 - - ------ + - + ------ 2 2 2 2
$$\left(\frac{5}{2} - \frac{\sqrt{21}}{2}\right) + \left(\frac{\sqrt{21}}{2} + \frac{5}{2}\right)$$
=
5
$$5$$
product
/ ____\ / ____\ |5 \/ 21 | |5 \/ 21 | |- - ------|*|- + ------| \2 2 / \2 2 /
$$\left(\frac{5}{2} - \frac{\sqrt{21}}{2}\right) \left(\frac{\sqrt{21}}{2} + \frac{5}{2}\right)$$
=
1
$$1$$
1
Rapid solution
[src]
____
5 \/ 21
x1 = - - ------
2 2
$$x_{1} = \frac{5}{2} - \frac{\sqrt{21}}{2}$$
____
5 \/ 21
x2 = - + ------
2 2
$$x_{2} = \frac{\sqrt{21}}{2} + \frac{5}{2}$$
x2 = sqrt(21)/2 + 5/2
The graph