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- Equation:
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Equation 10*(x-9)=7
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Equation 4(x-6)=x-9
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Equation x^2+11=0
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Equation k^2+k-6=0
- Express {x} in terms of y where:
- -11*x+1*y=12
- 18*x-12*y=-20
- 7*x-2*y=20
- -6*x-20*y=8
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Identical expressions
- √7x+ one -√ six -x=√ fifteen +2x
- √7x plus 1 minus √6 minus x equally √15 plus 2x
- √7x plus one minus √ six minus x equally √ fifteen plus 2x
-
Similar expressions
- √7x-1-√6-x=√15+2x
- √7x+1-√6-x=√15-2x
- √7x+1-√6+x=√15+2x
- √7x+1+√6-x=√15+2x
√7x+1-√6-x=√15+2x equation
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The solution
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_____ ___ ____ \/ 7*x + 1 - \/ 6 - x = \/ 15 + 2*x
$$- x + \left(\left(\sqrt{7 x} + 1\right) - \sqrt{6}\right) = 2 x + \sqrt{15}$$
Detail solution
Given the equation
$$- x + \left(\left(\sqrt{7 x} + 1\right) - \sqrt{6}\right) = 2 x + \sqrt{15}$$
Transfer the right side of the equation left part with negative sign
$$\sqrt{7} \sqrt{x} = 3 x - 1 + \sqrt{6} + \sqrt{15}$$
We raise the equation sides to 2-th degree
$$7 x = \left(3 x - 1 + \sqrt{6} + \sqrt{15}\right)^{2}$$
$$7 x = 9 x^{2} - 6 x + 6 \sqrt{6} x + 6 \sqrt{15} x - 2 \sqrt{15} - 2 \sqrt{6} + 6 \sqrt{10} + 22$$
Transfer the right side of the equation left part with negative sign
$$- 9 x^{2} - 6 \sqrt{15} x - 6 \sqrt{6} x + 13 x - 22 - 6 \sqrt{10} + 2 \sqrt{6} + 2 \sqrt{15} = 0$$
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
$$a = -9$$
$$b = - 6 \sqrt{15} - 6 \sqrt{6} + 13$$
$$c = -22 - 6 \sqrt{10} + 2 \sqrt{6} + 2 \sqrt{15}$$
, then
Because D<0, then the equation
has no real roots,
but complex roots is exists.
or
$$x_{1} = - \frac{\sqrt{15}}{3} - \frac{\sqrt{6}}{3} + \frac{13}{18} - \frac{\sqrt{-792 - 216 \sqrt{10} + 72 \sqrt{6} + 72 \sqrt{15} + \left(- 6 \sqrt{15} - 6 \sqrt{6} + 13\right)^{2}}}{18}$$
$$x_{2} = - \frac{\sqrt{15}}{3} - \frac{\sqrt{6}}{3} + \frac{13}{18} + \frac{\sqrt{-792 - 216 \sqrt{10} + 72 \sqrt{6} + 72 \sqrt{15} + \left(- 6 \sqrt{15} - 6 \sqrt{6} + 13\right)^{2}}}{18}$$
$$- x + \left(\left(\sqrt{7 x} + 1\right) - \sqrt{6}\right) = 2 x + \sqrt{15}$$
Transfer the right side of the equation left part with negative sign
$$\sqrt{7} \sqrt{x} = 3 x - 1 + \sqrt{6} + \sqrt{15}$$
We raise the equation sides to 2-th degree
$$7 x = \left(3 x - 1 + \sqrt{6} + \sqrt{15}\right)^{2}$$
$$7 x = 9 x^{2} - 6 x + 6 \sqrt{6} x + 6 \sqrt{15} x - 2 \sqrt{15} - 2 \sqrt{6} + 6 \sqrt{10} + 22$$
Transfer the right side of the equation left part with negative sign
$$- 9 x^{2} - 6 \sqrt{15} x - 6 \sqrt{6} x + 13 x - 22 - 6 \sqrt{10} + 2 \sqrt{6} + 2 \sqrt{15} = 0$$
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
$$a = -9$$
$$b = - 6 \sqrt{15} - 6 \sqrt{6} + 13$$
$$c = -22 - 6 \sqrt{10} + 2 \sqrt{6} + 2 \sqrt{15}$$
, then
D = b^2 - 4 * a * c =
(13 - 6*sqrt(6) - 6*sqrt(15))^2 - 4 * (-9) * (-22 - 6*sqrt(10) + 2*sqrt(6) + 2*sqrt(15)) = -792 + (13 - 6*sqrt(6) - 6*sqrt(15))^2 - 216*sqrt(10) + 72*sqrt(6) + 72*sqrt(15)
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
$$x_{1} = - \frac{\sqrt{15}}{3} - \frac{\sqrt{6}}{3} + \frac{13}{18} - \frac{\sqrt{-792 - 216 \sqrt{10} + 72 \sqrt{6} + 72 \sqrt{15} + \left(- 6 \sqrt{15} - 6 \sqrt{6} + 13\right)^{2}}}{18}$$
$$x_{2} = - \frac{\sqrt{15}}{3} - \frac{\sqrt{6}}{3} + \frac{13}{18} + \frac{\sqrt{-792 - 216 \sqrt{10} + 72 \sqrt{6} + 72 \sqrt{15} + \left(- 6 \sqrt{15} - 6 \sqrt{6} + 13\right)^{2}}}{18}$$
Sum and product of roots
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sum
_____________________________ _____________________________
___ ____ / ___ ____ ___ ____ / ___ ____
13 \/ 6 \/ 15 I*\/ -133 + 84*\/ 6 + 84*\/ 15 13 \/ 6 \/ 15 I*\/ -133 + 84*\/ 6 + 84*\/ 15
-- - ----- - ------ - ---------------------------------- + -- - ----- - ------ + ----------------------------------
18 3 3 18 18 3 3 18
$$\left(- \frac{\sqrt{15}}{3} - \frac{\sqrt{6}}{3} + \frac{13}{18} - \frac{i \sqrt{-133 + 84 \sqrt{6} + 84 \sqrt{15}}}{18}\right) + \left(- \frac{\sqrt{15}}{3} - \frac{\sqrt{6}}{3} + \frac{13}{18} + \frac{i \sqrt{-133 + 84 \sqrt{6} + 84 \sqrt{15}}}{18}\right)$$
=
___ ____ 13 2*\/ 6 2*\/ 15 -- - ------- - -------- 9 3 3
$$- \frac{2 \sqrt{15}}{3} - \frac{2 \sqrt{6}}{3} + \frac{13}{9}$$
product
/ _____________________________\ / _____________________________\ | ___ ____ / ___ ____ | | ___ ____ / ___ ____ | |13 \/ 6 \/ 15 I*\/ -133 + 84*\/ 6 + 84*\/ 15 | |13 \/ 6 \/ 15 I*\/ -133 + 84*\/ 6 + 84*\/ 15 | |-- - ----- - ------ - ----------------------------------|*|-- - ----- - ------ + ----------------------------------| \18 3 3 18 / \18 3 3 18 /
$$\left(- \frac{\sqrt{15}}{3} - \frac{\sqrt{6}}{3} + \frac{13}{18} - \frac{i \sqrt{-133 + 84 \sqrt{6} + 84 \sqrt{15}}}{18}\right) \left(- \frac{\sqrt{15}}{3} - \frac{\sqrt{6}}{3} + \frac{13}{18} + \frac{i \sqrt{-133 + 84 \sqrt{6} + 84 \sqrt{15}}}{18}\right)$$
=
___ ____ ____ 22 2*\/ 6 2*\/ 15 2*\/ 10 -- - ------- - -------- + -------- 9 9 9 3
$$- \frac{2 \sqrt{15}}{9} - \frac{2 \sqrt{6}}{9} + \frac{2 \sqrt{10}}{3} + \frac{22}{9}$$
22/9 - 2*sqrt(6)/9 - 2*sqrt(15)/9 + 2*sqrt(10)/3
Rapid solution
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_____________________________
___ ____ / ___ ____
13 \/ 6 \/ 15 I*\/ -133 + 84*\/ 6 + 84*\/ 15
x1 = -- - ----- - ------ - ----------------------------------
18 3 3 18
$$x_{1} = - \frac{\sqrt{15}}{3} - \frac{\sqrt{6}}{3} + \frac{13}{18} - \frac{i \sqrt{-133 + 84 \sqrt{6} + 84 \sqrt{15}}}{18}$$
_____________________________
___ ____ / ___ ____
13 \/ 6 \/ 15 I*\/ -133 + 84*\/ 6 + 84*\/ 15
x2 = -- - ----- - ------ + ----------------------------------
18 3 3 18
$$x_{2} = - \frac{\sqrt{15}}{3} - \frac{\sqrt{6}}{3} + \frac{13}{18} + \frac{i \sqrt{-133 + 84 \sqrt{6} + 84 \sqrt{15}}}{18}$$
x2 = -sqrt(15)/3 - sqrt(6)/3 + 13/18 + i*sqrt(-133 + 84*sqrt(6) + 84*sqrt(15))/18
Numerical answer
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x1 = -1.38526880744131 - 1.10845201185158*i
x2 = -1.38526880744131 + 1.10845201185158*i
x2 = -1.38526880744131 + 1.10845201185158*i