Mister Exam
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How to use it?
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Equation x^4-2*x^3+x^2-1=0
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Identical expressions
- six *x^ two + twenty *x+ nine = zero
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Similar expressions
- 6*x^2+20*x-9=0
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6*x^2+20*x+9=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
$$a = 6$$
$$b = 20$$
$$c = 9$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = - \frac{5}{3} + \frac{\sqrt{46}}{6}$$
$$x_{2} = - \frac{5}{3} - \frac{\sqrt{46}}{6}$$
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
$$a = 6$$
$$b = 20$$
$$c = 9$$
, then
D = b^2 - 4 * a * c =
(20)^2 - 4 * (6) * (9) = 184
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - \frac{5}{3} + \frac{\sqrt{46}}{6}$$
$$x_{2} = - \frac{5}{3} - \frac{\sqrt{46}}{6}$$
Vieta's Theorem
rewrite the equation
$$\left(6 x^{2} + 20 x\right) + 9 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{10 x}{3} + \frac{3}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{10}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{3}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{10}{3}$$
$$x_{1} x_{2} = \frac{3}{2}$$
$$\left(6 x^{2} + 20 x\right) + 9 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{10 x}{3} + \frac{3}{2} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{10}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{3}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{10}{3}$$
$$x_{1} x_{2} = \frac{3}{2}$$
Rapid solution
[src]
____
5 \/ 46
x1 = - - - ------
3 6
$$x_{1} = - \frac{5}{3} - \frac{\sqrt{46}}{6}$$
____
5 \/ 46
x2 = - - + ------
3 6
$$x_{2} = - \frac{5}{3} + \frac{\sqrt{46}}{6}$$
x2 = -5/3 + sqrt(46)/6
Sum and product of roots
[src]
sum
____ ____ 5 \/ 46 5 \/ 46 - - - ------ + - - + ------ 3 6 3 6
$$\left(- \frac{5}{3} - \frac{\sqrt{46}}{6}\right) + \left(- \frac{5}{3} + \frac{\sqrt{46}}{6}\right)$$
=
-10/3
$$- \frac{10}{3}$$
product
/ ____\ / ____\ | 5 \/ 46 | | 5 \/ 46 | |- - - ------|*|- - + ------| \ 3 6 / \ 3 6 /
$$\left(- \frac{5}{3} - \frac{\sqrt{46}}{6}\right) \left(- \frac{5}{3} + \frac{\sqrt{46}}{6}\right)$$
=
3/2
$$\frac{3}{2}$$
3/2