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Identical expressions
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Similar expressions
- 4x^3+3x^2
4x^3-3x^2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$4 x^{3} - 3 x^{2} = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(4 x^{2} - 3 x\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$4 x^{2} - 3 x = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 4$$
$$b = -3$$
$$c = 0$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{2} = \frac{3}{4}$$
Simplify
$$x_{3} = 0$$
Simplify
The final answer for (4*x^3 - 3*x^2) + 0 = 0:
$$x_{1} = 0$$
$$x_{2} = \frac{3}{4}$$
$$x_{3} = 0$$
$$4 x^{3} - 3 x^{2} = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(4 x^{2} - 3 x\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$4 x^{2} - 3 x = 0$$
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:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 4$$
$$b = -3$$
$$c = 0$$
, then
D = b^2 - 4 * a * c =
(-3)^2 - 4 * (4) * (0) = 9
Because D > 0, then the equation has two roots.
x2 = (-b + sqrt(D)) / (2*a)
x3 = (-b - sqrt(D)) / (2*a)
or
$$x_{2} = \frac{3}{4}$$
Simplify
$$x_{3} = 0$$
Simplify
The final answer for (4*x^3 - 3*x^2) + 0 = 0:
$$x_{1} = 0$$
$$x_{2} = \frac{3}{4}$$
$$x_{3} = 0$$
Vieta's Theorem
rewrite the equation
$$4 x^{3} - 3 x^{2} = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{3 x^{2}}{4} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{3}{4}$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{3}{4}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = 0$$
$$4 x^{3} - 3 x^{2} = 0$$
of
$$a x^{3} + b x^{2} + c x + d = 0$$
as reduced cubic equation
$$x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0$$
$$x^{3} - \frac{3 x^{2}}{4} = 0$$
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{3}{4}$$
$$q = \frac{c}{a}$$
$$q = 0$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = \frac{3}{4}$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 0$$
$$x_{1} x_{2} x_{3} = 0$$
Sum and product of roots
[src]
sum
0 + 0 + 3/4
$$\left(0 + 0\right) + \frac{3}{4}$$
=
3/4
$$\frac{3}{4}$$
product
1*0*3/4
$$1 \cdot 0 \cdot \frac{3}{4}$$
=
0
$$0$$
0
The graph