Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x+4=-1
-
Equation x-4=0
-
Equation e^x+1=0
-
Equation 3/5*y=29/10-1/5
- Express {x} in terms of y where:
- -11*x+16*y=3
- -19*x-15*y=15
- -4*x-13*y=-14
- 3*x-14*y=-2
- 2x^2+3y^2
-
Identical expressions
- two x^ two + three y^2=3
- 2x squared plus 3y squared equally 3
- two x to the power of two plus three y squared equally 3
- 2x2+3y2=3
- 2x²+3y²=3
- 2x to the power of 2+3y to the power of 2=3
-
Similar expressions
- 2x^2-3y^2=3
2x^2+3y^2=3 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
$$2 x^{2} + 3 y^{2} = 3$$
to
$$\left(2 x^{2} + 3 y^{2}\right) - 3 = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 2$$
$$b = 0$$
$$c = 3 y^{2} - 3$$
, then
$$D = b^2 - 4 * a * c = $$
$$- 2 \cdot 4 \cdot \left(3 y^{2} - 3\right) + 0^{2} = - 24 y^{2} + 24$$
The equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = \frac{\sqrt{- 24 y^{2} + 24}}{4}$$
Simplify
$$x_{2} = - \frac{\sqrt{- 24 y^{2} + 24}}{4}$$
Simplify
left part with negative sign.
The equation is transformed from
$$2 x^{2} + 3 y^{2} = 3$$
to
$$\left(2 x^{2} + 3 y^{2}\right) - 3 = 0$$
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 $D = b^2 - 4 a c$ is the discriminant.
Because
$$a = 2$$
$$b = 0$$
$$c = 3 y^{2} - 3$$
, then
$$D = b^2 - 4 * a * c = $$
$$- 2 \cdot 4 \cdot \left(3 y^{2} - 3\right) + 0^{2} = - 24 y^{2} + 24$$
The equation has two roots.
$$x_1 = \frac{(-b + \sqrt{D})}{2 a}$$
$$x_2 = \frac{(-b - \sqrt{D})}{2 a}$$
or
$$x_{1} = \frac{\sqrt{- 24 y^{2} + 24}}{4}$$
Simplify
$$x_{2} = - \frac{\sqrt{- 24 y^{2} + 24}}{4}$$
Simplify
Vieta's Theorem
rewrite the equation
$$2 x^{2} + 3 y^{2} = 3$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{3 y^{2}}{2} - \frac{3}{2} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{3 y^{2}}{2} - \frac{3}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = \frac{3 y^{2}}{2} - \frac{3}{2}$$
$$2 x^{2} + 3 y^{2} = 3$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} + \frac{3 y^{2}}{2} - \frac{3}{2} = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = \frac{3 y^{2}}{2} - \frac{3}{2}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = \frac{3 y^{2}}{2} - \frac{3}{2}$$
The graph
Sum and product of roots
[src]
sum
__________ __________
/ 2 / 2
-\/ 6 - 6*y \/ 6 - 6*y
--------------- + -------------
2 2
$$\left(- \frac{\sqrt{- 6 y^{2} + 6}}{2}\right) + \left(\frac{\sqrt{- 6 y^{2} + 6}}{2}\right)$$
=
0
$$0$$
product
__________ __________
/ 2 / 2
-\/ 6 - 6*y \/ 6 - 6*y
--------------- * -------------
2 2
$$\left(- \frac{\sqrt{- 6 y^{2} + 6}}{2}\right) * \left(\frac{\sqrt{- 6 y^{2} + 6}}{2}\right)$$
=
2 3 3*y - - + ---- 2 2
$$\frac{3 y^{2}}{2} - \frac{3}{2}$$