Mister Exam
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Identical expressions
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Similar expressions
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4x-3x^2+2=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = -3$$
$$b = 4$$
$$c = 2$$
, then
$$D = b^2 - 4\ a\ c = $$
$$4^{2} - \left(-3\right) 4 \cdot 2 = 40$$
Because D > 0, then the equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = - \frac{\sqrt{10}}{3} + \frac{2}{3}$$
Simplify
$$x_{2} = \frac{2}{3} + \frac{\sqrt{10}}{3}$$
Simplify
$$a\ x^2 + b\ x + c = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = -3$$
$$b = 4$$
$$c = 2$$
, then
$$D = b^2 - 4\ a\ c = $$
$$4^{2} - \left(-3\right) 4 \cdot 2 = 40$$
Because D > 0, then the equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = - \frac{\sqrt{10}}{3} + \frac{2}{3}$$
Simplify
$$x_{2} = \frac{2}{3} + \frac{\sqrt{10}}{3}$$
Simplify
Vieta's Theorem
rewrite the equation
$$- 3 x^{2} + 4 x + 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{4 x}{3} - \frac{2}{3} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{4}{3}$$
$$q = \frac{c}{a}$$
$$q = - \frac{2}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{4}{3}$$
$$x_{1} x_{2} = - \frac{2}{3}$$
$$- 3 x^{2} + 4 x + 2 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{4 x}{3} - \frac{2}{3} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{4}{3}$$
$$q = \frac{c}{a}$$
$$q = - \frac{2}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{4}{3}$$
$$x_{1} x_{2} = - \frac{2}{3}$$
Rapid solution
[src]
____
2 \/ 10
x_1 = - - ------
3 3
$$x_{1} = - \frac{\sqrt{10}}{3} + \frac{2}{3}$$
____
2 \/ 10
x_2 = - + ------
3 3
$$x_{2} = \frac{2}{3} + \frac{\sqrt{10}}{3}$$
Sum and product of roots
[src]
sum
____ ____ 2 \/ 10 2 \/ 10 - - ------ + - + ------ 3 3 3 3
$$\left(- \frac{\sqrt{10}}{3} + \frac{2}{3}\right) + \left(\frac{2}{3} + \frac{\sqrt{10}}{3}\right)$$
=
4/3
$$\frac{4}{3}$$
product
____ ____ 2 \/ 10 2 \/ 10 - - ------ * - + ------ 3 3 3 3
$$\left(- \frac{\sqrt{10}}{3} + \frac{2}{3}\right) * \left(\frac{2}{3} + \frac{\sqrt{10}}{3}\right)$$
=
-2/3
$$- \frac{2}{3}$$
The graph