Mister Exam
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How to use it?
- Equation:
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Equation x^3+2*x^2-9*x-18=0
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Equation 5*x^2=35*x
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Equation 4*(x-8)=-5
-
Equation 4=7-6(5x-1)
- Express {x} in terms of y where:
- 8*x-16*y=14
- -17*x-8*y=7
- -3*x-20*y=-13
- 15*x-3*y=19
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Identical expressions
- (4x- one) two -2x(8x- three)= three
- (4x minus 1)2 minus 2x(8x minus 3) equally 3
- (4x minus one) two minus 2x(8x minus three) equally three
- 4x-12-2x8x-3=3
-
Similar expressions
- (4x-1)2+2x(8x-3)=3
- (4x+1)2-2x(8x-3)=3
- (4x-1)2-2x(8x+3)=3
(4x-1)2-2x(8x-3)=3 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
(4*x - 1)*2 - 2*x*(8*x - 3) = 3
$$- 2 x \left(8 x - 3\right) + 2 \left(4 x - 1\right) = 3$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$- 2 x \left(8 x - 3\right) + 2 \left(4 x - 1\right) = 3$$
to
$$\left(- 2 x \left(8 x - 3\right) + 2 \left(4 x - 1\right)\right) - 3 = 0$$
Expand the expression in the equation
$$\left(- 2 x \left(8 x - 3\right) + 2 \left(4 x - 1\right)\right) - 3 = 0$$
We get the quadratic equation
$$- 16 x^{2} + 14 x - 5 = 0$$
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
$$a = -16$$
$$b = 14$$
$$c = -5$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = \frac{7}{16} - \frac{\sqrt{31} i}{16}$$
$$x_{2} = \frac{7}{16} + \frac{\sqrt{31} i}{16}$$
left part with negative sign.
The equation is transformed from
$$- 2 x \left(8 x - 3\right) + 2 \left(4 x - 1\right) = 3$$
to
$$\left(- 2 x \left(8 x - 3\right) + 2 \left(4 x - 1\right)\right) - 3 = 0$$
Expand the expression in the equation
$$\left(- 2 x \left(8 x - 3\right) + 2 \left(4 x - 1\right)\right) - 3 = 0$$
We get the quadratic equation
$$- 16 x^{2} + 14 x - 5 = 0$$
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
$$a = -16$$
$$b = 14$$
$$c = -5$$
, then
D = b^2 - 4 * a * c =
(14)^2 - 4 * (-16) * (-5) = -124
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{7}{16} - \frac{\sqrt{31} i}{16}$$
$$x_{2} = \frac{7}{16} + \frac{\sqrt{31} i}{16}$$
Rapid solution
[src]
____
7 I*\/ 31
x1 = -- - --------
16 16
$$x_{1} = \frac{7}{16} - \frac{\sqrt{31} i}{16}$$
____
7 I*\/ 31
x2 = -- + --------
16 16
$$x_{2} = \frac{7}{16} + \frac{\sqrt{31} i}{16}$$
x2 = 7/16 + sqrt(31)*i/16
Sum and product of roots
[src]
sum
____ ____ 7 I*\/ 31 7 I*\/ 31 -- - -------- + -- + -------- 16 16 16 16
$$\left(\frac{7}{16} - \frac{\sqrt{31} i}{16}\right) + \left(\frac{7}{16} + \frac{\sqrt{31} i}{16}\right)$$
=
7/8
$$\frac{7}{8}$$
product
/ ____\ / ____\ |7 I*\/ 31 | |7 I*\/ 31 | |-- - --------|*|-- + --------| \16 16 / \16 16 /
$$\left(\frac{7}{16} - \frac{\sqrt{31} i}{16}\right) \left(\frac{7}{16} + \frac{\sqrt{31} i}{16}\right)$$
=
5/16
$$\frac{5}{16}$$
5/16
Numerical answer
[src]
x1 = 0.4375 + 0.347985272676876*i
x2 = 0.4375 - 0.347985272676876*i
x2 = 0.4375 - 0.347985272676876*i