Mister Exam
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How to use it?
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Identical expressions
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Similar expressions
- 1,5*x^2+6*x+2=6
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1,5*x^2-6*x+2=6 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\left(\frac{3 x^{2}}{2} - 6 x\right) + 2 = 6$$
to
$$\left(\left(\frac{3 x^{2}}{2} - 6 x\right) + 2\right) - 6 = 0$$
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
$$a = \frac{3}{2}$$
$$b = -6$$
$$c = -4$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = 2 + \frac{2 \sqrt{15}}{3}$$
$$x_{2} = 2 - \frac{2 \sqrt{15}}{3}$$
left part with negative sign.
The equation is transformed from
$$\left(\frac{3 x^{2}}{2} - 6 x\right) + 2 = 6$$
to
$$\left(\left(\frac{3 x^{2}}{2} - 6 x\right) + 2\right) - 6 = 0$$
This equation is of the form
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
$$a = \frac{3}{2}$$
$$b = -6$$
$$c = -4$$
, then
D = b^2 - 4 * a * c =
(-6)^2 - 4 * (3/2) * (-4) = 60
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = 2 + \frac{2 \sqrt{15}}{3}$$
$$x_{2} = 2 - \frac{2 \sqrt{15}}{3}$$
Vieta's Theorem
rewrite the equation
$$\left(\frac{3 x^{2}}{2} - 6 x\right) + 2 = 6$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 4 x - \frac{8}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = - \frac{8}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = - \frac{8}{3}$$
$$\left(\frac{3 x^{2}}{2} - 6 x\right) + 2 = 6$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 4 x - \frac{8}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = - \frac{8}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = - \frac{8}{3}$$
Sum and product of roots
[src]
sum
____ ____
2*\/ 15 2*\/ 15
2 - -------- + 2 + --------
3 3
$$\left(2 - \frac{2 \sqrt{15}}{3}\right) + \left(2 + \frac{2 \sqrt{15}}{3}\right)$$
=
4
$$4$$
product
/ ____\ / ____\ | 2*\/ 15 | | 2*\/ 15 | |2 - --------|*|2 + --------| \ 3 / \ 3 /
$$\left(2 - \frac{2 \sqrt{15}}{3}\right) \left(2 + \frac{2 \sqrt{15}}{3}\right)$$
=
-8/3
$$- \frac{8}{3}$$
-8/3
Rapid solution
[src]
____
2*\/ 15
x1 = 2 - --------
3
$$x_{1} = 2 - \frac{2 \sqrt{15}}{3}$$
____
2*\/ 15
x2 = 2 + --------
3
$$x_{2} = 2 + \frac{2 \sqrt{15}}{3}$$
x2 = 2 + 2*sqrt(15)/3