Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 12-x=x
-
Equation 4x+5=2x-7
-
Equation 3*x^2-9=0
-
Equation x^2-x+1=0
- Express {x} in terms of y where:
- 20*x+9*y=17
- 11*x-11*y=-17
- -17*x-12*y=-9
- -6*x+1*y=-5
-
Identical expressions
- 13x-3x^ two =- fourteen
- 13x minus 3x squared equally minus 14
- 13x minus 3x to the power of two equally minus fourteen
- 13x-3x2=-14
- 13x-3x²=-14
- 13x-3x to the power of 2=-14
-
Similar expressions
- x^2*(2-2*x)/(x-1)^4+2*x/(x-1)^2=0
- 13x-3x^2=+14
- 13x+3x^2=-14
13x-3x^2=-14 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
$$- 3 x^{2} + 13 x = -14$$
to
$$\left(- 3 x^{2} + 13 x\right) + 14 = 0$$
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
$$a = -3$$
$$b = 13$$
$$c = 14$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{13}{6} - \frac{\sqrt{337}}{6}$$
$$x_{2} = \frac{13}{6} + \frac{\sqrt{337}}{6}$$
left part with negative sign.
The equation is transformed from
$$- 3 x^{2} + 13 x = -14$$
to
$$\left(- 3 x^{2} + 13 x\right) + 14 = 0$$
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
$$a = -3$$
$$b = 13$$
$$c = 14$$
, then
D = b^2 - 4 * a * c =
(13)^2 - 4 * (-3) * (14) = 337
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{13}{6} - \frac{\sqrt{337}}{6}$$
$$x_{2} = \frac{13}{6} + \frac{\sqrt{337}}{6}$$
Vieta's Theorem
rewrite the equation
$$- 3 x^{2} + 13 x = -14$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{13 x}{3} - \frac{14}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{13}{3}$$
$$q = \frac{c}{a}$$
$$q = - \frac{14}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{13}{3}$$
$$x_{1} x_{2} = - \frac{14}{3}$$
$$- 3 x^{2} + 13 x = -14$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{13 x}{3} - \frac{14}{3} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{13}{3}$$
$$q = \frac{c}{a}$$
$$q = - \frac{14}{3}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{13}{3}$$
$$x_{1} x_{2} = - \frac{14}{3}$$
Sum and product of roots
[src]
sum
_____ _____ 13 \/ 337 13 \/ 337 -- - ------- + -- + ------- 6 6 6 6
$$\left(\frac{13}{6} - \frac{\sqrt{337}}{6}\right) + \left(\frac{13}{6} + \frac{\sqrt{337}}{6}\right)$$
=
13/3
$$\frac{13}{3}$$
product
/ _____\ / _____\ |13 \/ 337 | |13 \/ 337 | |-- - -------|*|-- + -------| \6 6 / \6 6 /
$$\left(\frac{13}{6} - \frac{\sqrt{337}}{6}\right) \left(\frac{13}{6} + \frac{\sqrt{337}}{6}\right)$$
=
-14/3
$$- \frac{14}{3}$$
-14/3
Rapid solution
[src]
_____
13 \/ 337
x1 = -- - -------
6 6
$$x_{1} = \frac{13}{6} - \frac{\sqrt{337}}{6}$$
_____
13 \/ 337
x2 = -- + -------
6 6
$$x_{2} = \frac{13}{6} + \frac{\sqrt{337}}{6}$$
x2 = 13/6 + sqrt(337)/6