Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation -x/7+x=15/7
-
Equation -33-y=-19
-
Equation x^2=2x+8
-
Equation 6=8-5(7x-1)
- Express {x} in terms of y where:
- -2*x+18*y=13
- -2*x-15*y=-1
- 5*x+4*y=-12
- 10*x-2*y=11
- Integral of d{x}:
-
12x
- The double integral of:
- 12x
- Derivative of:
-
12x
-
Identical expressions
- 1 two x=- thirty-six +3x^2
- 12x equally minus 36 plus 3x squared
- 1 two x equally minus thirty minus six plus 3x squared
- 12x=-36+3x2
- 12x=-36+3x²
- 12x=-36+3x to the power of 2
-
Similar expressions
- 5*x^2-12*x=0
- 2*(x-1)^4-5*(x-1)^2*(x-2)^2+2*(x-2)^4=0
- x^2-12*x+36=0
- 12x=+36+3x^2
- 12x=-36-3x^2
12x=-36+3x^2 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
$$12 x = 3 x^{2} - 36$$
to
$$12 x - \left(3 x^{2} - 36\right) = 0$$
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
$$a = -3$$
$$b = 12$$
$$c = 36$$
, then
$$D = b^2 - 4 * a * c = $$
$$12^{2} - \left(-3\right) 4 \cdot 36 = 576$$
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
$$x_{1} = -2$$
Simplify
$$x_{2} = 6$$
Simplify
left part with negative sign.
The equation is transformed from
$$12 x = 3 x^{2} - 36$$
to
$$12 x - \left(3 x^{2} - 36\right) = 0$$
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
$$a = -3$$
$$b = 12$$
$$c = 36$$
, then
$$D = b^2 - 4 * a * c = $$
$$12^{2} - \left(-3\right) 4 \cdot 36 = 576$$
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
$$x_{1} = -2$$
Simplify
$$x_{2} = 6$$
Simplify
Vieta's Theorem
rewrite the equation
$$12 x = 3 x^{2} - 36$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 4 x - 12 = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = -12$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = -12$$
$$12 x = 3 x^{2} - 36$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 4 x - 12 = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -4$$
$$q = \frac{c}{a}$$
$$q = -12$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 4$$
$$x_{1} x_{2} = -12$$
Sum and product of roots
[src]
sum
-2 + 6
$$\left(-2\right) + \left(6\right)$$
=
4
$$4$$
product
-2 * 6
$$\left(-2\right) * \left(6\right)$$
=
-12
$$-12$$
The graph