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- La ecuación:
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Ecuación 5*x^2+2*x-7=0
-
Ecuación 12,4-x=16
-
Ecuación x+120=40*5
-
Ecuación y-7/12=4
- Expresar {x} en función de y en la ecuación:
- -18*x-8*y=-18
- -13*x+12*y=-19
- -1*x+11*y=-9
- 16*x-11*y=13
-
Expresiones idénticas
- (10a+b)(a+b)= cuatrocientos cinco
- (10a más b)(a más b) es igual a 405
- (10a más b)(a más b) es igual a cuatrocientos cinco
- 10a+ba+b=405
-
Expresiones semejantes
- x+120=40*5
- (10a+b)(a-b)=405
- (10a-b)(a+b)=405
- (x-40)*5/x+(x-40)*5/(x+50)=1
(10a+b)(a+b)=405 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$\left(a + b\right) \left(10 a + b\right) = 405$$
en
$$\left(a + b\right) \left(10 a + b\right) - 405 = 0$$
Abramos la expresión en la ecuación
$$\left(a + b\right) \left(10 a + b\right) - 405 = 0$$
Obtenemos la ecuación cuadrática
$$10 a^{2} + 11 a b + b^{2} - 405 = 0$$
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$b_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$b_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = 11 a$$
$$c = 10 a^{2} - 405$$
, entonces
La ecuación tiene dos raíces.
o
$$b_{1} = - \frac{11 a}{2} + \frac{\sqrt{81 a^{2} + 1620}}{2}$$
$$b_{2} = - \frac{11 a}{2} - \frac{\sqrt{81 a^{2} + 1620}}{2}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$\left(a + b\right) \left(10 a + b\right) = 405$$
en
$$\left(a + b\right) \left(10 a + b\right) - 405 = 0$$
Abramos la expresión en la ecuación
$$\left(a + b\right) \left(10 a + b\right) - 405 = 0$$
Obtenemos la ecuación cuadrática
$$10 a^{2} + 11 a b + b^{2} - 405 = 0$$
Es la ecuación de la forma
a*b^2 + b*b + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$b_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$b_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 1$$
$$b = 11 a$$
$$c = 10 a^{2} - 405$$
, entonces
D = b^2 - 4 * a * c =
(11*a)^2 - 4 * (1) * (-405 + 10*a^2) = 1620 + 81*a^2
La ecuación tiene dos raíces.
b1 = (-b + sqrt(D)) / (2*a)
b2 = (-b - sqrt(D)) / (2*a)
o
$$b_{1} = - \frac{11 a}{2} + \frac{\sqrt{81 a^{2} + 1620}}{2}$$
$$b_{2} = - \frac{11 a}{2} - \frac{\sqrt{81 a^{2} + 1620}}{2}$$
Gráfica
Suma y producto de raíces
[src]
suma
/ ___________________________________________ \ ___________________________________________ / ___________________________________________ \ ___________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\ | / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), 20 + re (a) - im (a)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), 20 + re (a) - im (a)/| | 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), 20 + re (a) - im (a)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), 20 + re (a) - im (a)/|
| 9*\/ \20 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------|| 9*\/ \20 + re (a) - im (a)/ + 4*im (a)*re (a) *cos|------------------------------------------| | 9*\/ \20 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------|| 9*\/ \20 + re (a) - im (a)/ + 4*im (a)*re (a) *cos|------------------------------------------|
11*re(a) | 11*im(a) \ 2 /| \ 2 / 11*re(a) | 11*im(a) \ 2 /| \ 2 /
- -------- + I*|- -------- - -------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------- + - -------- + I*|- -------- + -------------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------
2 \ 2 2 / 2 2 \ 2 2 / 2
$$\left(i \left(- \frac{9 \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} - \frac{11 \operatorname{im}{\left(a\right)}}{2}\right) - \frac{9 \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} - \frac{11 \operatorname{re}{\left(a\right)}}{2}\right) + \left(i \left(\frac{9 \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} - \frac{11 \operatorname{im}{\left(a\right)}}{2}\right) + \frac{9 \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} - \frac{11 \operatorname{re}{\left(a\right)}}{2}\right)$$
=
/ ___________________________________________ \ / ___________________________________________ \
| / 2 / / 2 2 \\| | / 2 / / 2 2 \\|
| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), 20 + re (a) - im (a)/|| | 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), 20 + re (a) - im (a)/||
| 9*\/ \20 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------|| | 9*\/ \20 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------||
| 11*im(a) \ 2 /| | 11*im(a) \ 2 /|
-11*re(a) + I*|- -------- - -------------------------------------------------------------------------------------------------| + I*|- -------- + -------------------------------------------------------------------------------------------------|
\ 2 2 / \ 2 2 /
$$i \left(- \frac{9 \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} - \frac{11 \operatorname{im}{\left(a\right)}}{2}\right) + i \left(\frac{9 \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} - \frac{11 \operatorname{im}{\left(a\right)}}{2}\right) - 11 \operatorname{re}{\left(a\right)}$$
producto
/ / ___________________________________________ \ ___________________________________________ \ / / ___________________________________________ \ ___________________________________________ \ | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), 20 + re (a) - im (a)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), 20 + re (a) - im (a)/|| | | 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), 20 + re (a) - im (a)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), 20 + re (a) - im (a)/|| | | 9*\/ \20 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------|| 9*\/ \20 + re (a) - im (a)/ + 4*im (a)*re (a) *cos|------------------------------------------|| | | 9*\/ \20 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------|| 9*\/ \20 + re (a) - im (a)/ + 4*im (a)*re (a) *cos|------------------------------------------|| | 11*re(a) | 11*im(a) \ 2 /| \ 2 /| | 11*re(a) | 11*im(a) \ 2 /| \ 2 /| |- -------- + I*|- -------- - -------------------------------------------------------------------------------------------------| - -------------------------------------------------------------------------------------------------|*|- -------- + I*|- -------- + -------------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------| \ 2 \ 2 2 / 2 / \ 2 \ 2 2 / 2 /
$$\left(i \left(- \frac{9 \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} - \frac{11 \operatorname{im}{\left(a\right)}}{2}\right) - \frac{9 \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} - \frac{11 \operatorname{re}{\left(a\right)}}{2}\right) \left(i \left(\frac{9 \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} - \frac{11 \operatorname{im}{\left(a\right)}}{2}\right) + \frac{9 \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} - \frac{11 \operatorname{re}{\left(a\right)}}{2}\right)$$
=
2 2 -405 - 10*im (a) + 10*re (a) + 20*I*im(a)*re(a)
$$10 \left(\operatorname{re}{\left(a\right)}\right)^{2} + 20 i \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - 10 \left(\operatorname{im}{\left(a\right)}\right)^{2} - 405$$
-405 - 10*im(a)^2 + 10*re(a)^2 + 20*i*im(a)*re(a)
Respuesta rápida
[src]
/ ___________________________________________ \ ___________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), 20 + re (a) - im (a)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), 20 + re (a) - im (a)/|
| 9*\/ \20 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------|| 9*\/ \20 + re (a) - im (a)/ + 4*im (a)*re (a) *cos|------------------------------------------|
11*re(a) | 11*im(a) \ 2 /| \ 2 /
b1 = - -------- + I*|- -------- - -------------------------------------------------------------------------------------------------| - -------------------------------------------------------------------------------------------------
2 \ 2 2 / 2
$$b_{1} = i \left(- \frac{9 \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} - \frac{11 \operatorname{im}{\left(a\right)}}{2}\right) - \frac{9 \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} - \frac{11 \operatorname{re}{\left(a\right)}}{2}$$
/ ___________________________________________ \ ___________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), 20 + re (a) - im (a)/|| 4 / / 2 2 \ 2 2 |atan2\2*im(a)*re(a), 20 + re (a) - im (a)/|
| 9*\/ \20 + re (a) - im (a)/ + 4*im (a)*re (a) *sin|------------------------------------------|| 9*\/ \20 + re (a) - im (a)/ + 4*im (a)*re (a) *cos|------------------------------------------|
11*re(a) | 11*im(a) \ 2 /| \ 2 /
b2 = - -------- + I*|- -------- + -------------------------------------------------------------------------------------------------| + -------------------------------------------------------------------------------------------------
2 \ 2 2 / 2
$$b_{2} = i \left(\frac{9 \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} - \frac{11 \operatorname{im}{\left(a\right)}}{2}\right) + \frac{9 \sqrt[4]{\left(\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},\left(\operatorname{re}{\left(a\right)}\right)^{2} - \left(\operatorname{im}{\left(a\right)}\right)^{2} + 20 \right)}}{2} \right)}}{2} - \frac{11 \operatorname{re}{\left(a\right)}}{2}$$
b2 = i*(9*((re(a)^2 - im(a)^2 + 20)^2 + 4*re(a)^2*im(a)^2)^(1/4)*sin(atan2(2*re(a)*im(a, re(a)^2 - im(a)^2 + 20)/2)/2 - 11*im(a)/2) + 9*((re(a)^2 - im(a)^2 + 20)^2 + 4*re(a)^2*im(a)^2)^(1/4)*cos(atan2(2*re(a)*im(a), re(a)^2 - im(a)^2 + 20)/2)/2 - 11*re(a)/2)