Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation sin(x)+1/2=0
-
Equation -15-3x=-7x+45
-
Equation 5x^2=-80+40x
-
Equation 6*x=0
- Express {x} in terms of y where:
- 17*x-19*y=1
- 4*x+13*y=13
- -16*x+14*y=-9
- -14*x+19*y=-3
- Graphing y =:
- 5x^2
- Canonical form:
- 5x^2
- Integral of d{x}:
-
5x^2
-
Identical expressions
- 5x^ two =- eighty +40x
- 5x squared equally minus 80 plus 40x
- 5x to the power of two equally minus eighty plus 40x
- 5x2=-80+40x
- 5x²=-80+40x
- 5x to the power of 2=-80+40x
-
Similar expressions
- x^3-5x^2-x+5=0
- 5x^2=-80-40x
- x^4-5x^2-6=0
- x^4+15x^2-16=0
- 5x^2=+80+40x
5x^2=-80+40x 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
$$5 x^{2} = 40 x - 80$$
to
$$5 x^{2} - \left(40 x - 80\right) = 0$$
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
$$a = 5$$
$$b = -40$$
$$c = 80$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-1\right) 5 \cdot 4 \cdot 80 + \left(-40\right)^{2} = 0$$
Because D = 0, then the equation has one root.
$$x_{1} = 4$$
left part with negative sign.
The equation is transformed from
$$5 x^{2} = 40 x - 80$$
to
$$5 x^{2} - \left(40 x - 80\right) = 0$$
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
$$a = 5$$
$$b = -40$$
$$c = 80$$
, then
$$D = b^2 - 4\ a\ c = $$
$$\left(-1\right) 5 \cdot 4 \cdot 80 + \left(-40\right)^{2} = 0$$
Because D = 0, then the equation has one root.
x = -b/2a = --40/2/(5)
$$x_{1} = 4$$
Vieta's Theorem
rewrite the equation
$$5 x^{2} = 40 x - 80$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 8 x + 16 = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -8$$
$$q = \frac{c}{a}$$
$$q = 16$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 8$$
$$x_{1} x_{2} = 16$$
$$5 x^{2} = 40 x - 80$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - 8 x + 16 = 0$$
$$p x + x^{2} + q = 0$$
where
$$p = \frac{b}{a}$$
$$p = -8$$
$$q = \frac{c}{a}$$
$$q = 16$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 8$$
$$x_{1} x_{2} = 16$$
Sum and product of roots
[src]
sum
4
$$\left(4\right)$$
=
4
$$4$$
product
4
$$\left(4\right)$$
=
4
$$4$$
The graph