Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 12/(x+5)=-12/5
-
Equation 2x^2-8=0
-
Equation x+5=0
-
Equation x^4+x^3+x^2+x+1=0
- Express {x} in terms of y where:
- -11*x+16*y=-5
- -3*x+3*y=15
- -16*x+3*y=-14
- -16*x-13*y=-1
- Integral of d{x}:
-
1÷x
- Graphing y =:
- 1÷x
- Derivative of:
-
1÷x
-
Identical expressions
- - three *x+ fifteen = one ÷x
- minus 3 multiply by x plus 15 equally 1÷x
- minus three multiply by x plus fifteen equally one ÷x
- -3x+15=1÷x
-
Similar expressions
- 2-3x+15-10=5x-1
- 6÷(3x+6)+(x+1÷(x+3))=2
- -3x^+15=0
- -3*x-15=1÷x
- 0,64=1÷(x-7,0375)
- 3*x+15=1÷x
-3*x+15=1÷x equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation:
$$15 - 3 x = \frac{1}{x}$$
Multiply the equation sides by the denominators:
and x
we get:
$$x \left(15 - 3 x\right) = \frac{x}{x}$$
$$- 3 x^{2} + 15 x = 1$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$- 3 x^{2} + 15 x = 1$$
to
$$- 3 x^{2} + 15 x - 1 = 0$$
This equation is of the form
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 - it is the discriminant.
Because
$$a = -3$$
$$b = 15$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{5}{2} - \frac{\sqrt{213}}{6}$$
$$x_{2} = \frac{\sqrt{213}}{6} + \frac{5}{2}$$
$$15 - 3 x = \frac{1}{x}$$
Multiply the equation sides by the denominators:
and x
we get:
$$x \left(15 - 3 x\right) = \frac{x}{x}$$
$$- 3 x^{2} + 15 x = 1$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$- 3 x^{2} + 15 x = 1$$
to
$$- 3 x^{2} + 15 x - 1 = 0$$
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 - it is the discriminant.
Because
$$a = -3$$
$$b = 15$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(15)^2 - 4 * (-3) * (-1) = 213
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{5}{2} - \frac{\sqrt{213}}{6}$$
$$x_{2} = \frac{\sqrt{213}}{6} + \frac{5}{2}$$
Rapid solution
[src]
_____
5 \/ 213
x1 = - - -------
2 6
$$x_{1} = \frac{5}{2} - \frac{\sqrt{213}}{6}$$
_____
5 \/ 213
x2 = - + -------
2 6
$$x_{2} = \frac{\sqrt{213}}{6} + \frac{5}{2}$$
x2 = sqrt(213)/6 + 5/2
Sum and product of roots
[src]
sum
_____ _____ 5 \/ 213 5 \/ 213 - - ------- + - + ------- 2 6 2 6
$$\left(\frac{5}{2} - \frac{\sqrt{213}}{6}\right) + \left(\frac{\sqrt{213}}{6} + \frac{5}{2}\right)$$
=
5
$$5$$
product
/ _____\ / _____\ |5 \/ 213 | |5 \/ 213 | |- - -------|*|- + -------| \2 6 / \2 6 /
$$\left(\frac{5}{2} - \frac{\sqrt{213}}{6}\right) \left(\frac{\sqrt{213}}{6} + \frac{5}{2}\right)$$
=
1/3
$$\frac{1}{3}$$
1/3