Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation exp(x)=0
-
Equation x^3-3x^2=0
-
Equation 0=2-3x
-
Equation 3x-7=x-11
- Express {x} in terms of y where:
- -17*x-12*y=-4
- 9*x-19*y=-20
- 15*x+17*y=-10
- -4*x+14*y=-4
- Derivative of:
- 5x+12
-
Identical expressions
- four (3x- two)2x=5x+ twelve
- 4(3x minus 2)2x equally 5x plus 12
- four (3x minus two)2x equally 5x plus twelve
- 43x-22x=5x+12
-
Similar expressions
- (5x+1)^2-(5c+1)(5x+2)=7
- 4(3x+2)2x=5x+12
- 5*x+12=8x+30
- 4(3x-2)2x=5x-12
- 3*x^2-15*x+125=17
- 2*x^2-5*x+12=25*x^2-20*x^3+4*x^4
4(3x-2)2x=5x+12 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
$$x 2 \cdot 4 \left(3 x - 2\right) = 5 x + 12$$
to
$$x 2 \cdot 4 \left(3 x - 2\right) + \left(- 5 x - 12\right) = 0$$
Expand the expression in the equation
$$x 2 \cdot 4 \left(3 x - 2\right) + \left(- 5 x - 12\right) = 0$$
We get the quadratic equation
$$24 x^{2} - 21 x - 12 = 0$$
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
$$a = 24$$
$$b = -21$$
$$c = -12$$
, then
Because D > 0, then the equation has two roots.
or
$$x_{1} = \frac{7}{16} + \frac{\sqrt{177}}{16}$$
$$x_{2} = \frac{7}{16} - \frac{\sqrt{177}}{16}$$
left part with negative sign.
The equation is transformed from
$$x 2 \cdot 4 \left(3 x - 2\right) = 5 x + 12$$
to
$$x 2 \cdot 4 \left(3 x - 2\right) + \left(- 5 x - 12\right) = 0$$
Expand the expression in the equation
$$x 2 \cdot 4 \left(3 x - 2\right) + \left(- 5 x - 12\right) = 0$$
We get the quadratic equation
$$24 x^{2} - 21 x - 12 = 0$$
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
$$a = 24$$
$$b = -21$$
$$c = -12$$
, then
D = b^2 - 4 * a * c =
(-21)^2 - 4 * (24) * (-12) = 1593
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{7}{16} + \frac{\sqrt{177}}{16}$$
$$x_{2} = \frac{7}{16} - \frac{\sqrt{177}}{16}$$
Sum and product of roots
[src]
sum
_____ _____ 7 \/ 177 7 \/ 177 -- - ------- + -- + ------- 16 16 16 16
$$\left(\frac{7}{16} - \frac{\sqrt{177}}{16}\right) + \left(\frac{7}{16} + \frac{\sqrt{177}}{16}\right)$$
=
7/8
$$\frac{7}{8}$$
product
/ _____\ / _____\ |7 \/ 177 | |7 \/ 177 | |-- - -------|*|-- + -------| \16 16 / \16 16 /
$$\left(\frac{7}{16} - \frac{\sqrt{177}}{16}\right) \left(\frac{7}{16} + \frac{\sqrt{177}}{16}\right)$$
=
-1/2
$$- \frac{1}{2}$$
-1/2
Rapid solution
[src]
_____
7 \/ 177
x1 = -- - -------
16 16
$$x_{1} = \frac{7}{16} - \frac{\sqrt{177}}{16}$$
_____
7 \/ 177
x2 = -- + -------
16 16
$$x_{2} = \frac{7}{16} + \frac{\sqrt{177}}{16}$$
x2 = 7/16 + sqrt(177)/16