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Identical expressions
- (one / thirty-eight)*((nine + five *y)/ fourteen)^ two +(y^ two)/ one hundred = one
- (1 divide by 38) multiply by ((9 plus 5 multiply by y) divide by 14) squared plus (y squared ) divide by 100 equally 1
- (one divide by thirty minus eight) multiply by ((nine plus five multiply by y) divide by fourteen) to the power of two plus (y to the power of two) divide by one hundred equally one
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- (1 divide by 38)*((9+5*y) divide by 14)^2+(y^2) divide by 100=1
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Similar expressions
- (1/38)*((9-5*y)/14)^2+(y^2)/100=1
- (1/38)*((9+5*y)/14)^2-(y^2)/100=1
(1/38)*((9+5*y)/14)^2+(y^2)/100=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2
/9 + 5*y\
|-------| 2
\ 14 / y
---------- + --- = 1
38 100
$$\frac{y^{2}}{100} + \frac{\left(\frac{5 y + 9}{14}\right)^{2}}{38} = 1$$
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$\frac{y^{2}}{100} + \frac{\left(\frac{5 y + 9}{14}\right)^{2}}{38} = 1$$
to
$$\left(\frac{y^{2}}{100} + \frac{\left(\frac{5 y + 9}{14}\right)^{2}}{38}\right) - 1 = 0$$
Expand the expression in the equation
$$\left(\frac{y^{2}}{100} + \frac{\left(\frac{5 y + 9}{14}\right)^{2}}{38}\right) - 1 = 0$$
We get the quadratic equation
$$\frac{2487 y^{2}}{186200} + \frac{45 y}{3724} - \frac{7367}{7448} = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = \frac{2487}{186200}$$
$$b = \frac{45}{3724}$$
$$c = - \frac{7367}{7448}$$
, then
Because D > 0, then the equation has two roots.
or
$$y_{1} = - \frac{375}{829} + \frac{35 \sqrt{374946}}{2487}$$
$$y_{2} = - \frac{35 \sqrt{374946}}{2487} - \frac{375}{829}$$
left part with negative sign.
The equation is transformed from
$$\frac{y^{2}}{100} + \frac{\left(\frac{5 y + 9}{14}\right)^{2}}{38} = 1$$
to
$$\left(\frac{y^{2}}{100} + \frac{\left(\frac{5 y + 9}{14}\right)^{2}}{38}\right) - 1 = 0$$
Expand the expression in the equation
$$\left(\frac{y^{2}}{100} + \frac{\left(\frac{5 y + 9}{14}\right)^{2}}{38}\right) - 1 = 0$$
We get the quadratic equation
$$\frac{2487 y^{2}}{186200} + \frac{45 y}{3724} - \frac{7367}{7448} = 0$$
This equation is of the form
a*y^2 + b*y + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = \frac{2487}{186200}$$
$$b = \frac{45}{3724}$$
$$c = - \frac{7367}{7448}$$
, then
D = b^2 - 4 * a * c =
(45/3724)^2 - 4 * (2487/186200) * (-7367/7448) = 9867/186200
Because D > 0, then the equation has two roots.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
or
$$y_{1} = - \frac{375}{829} + \frac{35 \sqrt{374946}}{2487}$$
$$y_{2} = - \frac{35 \sqrt{374946}}{2487} - \frac{375}{829}$$
Rapid solution
[src]
________
375 35*\/ 374946
y1 = - --- + -------------
829 2487
$$y_{1} = - \frac{375}{829} + \frac{35 \sqrt{374946}}{2487}$$
________
375 35*\/ 374946
y2 = - --- - -------------
829 2487
$$y_{2} = - \frac{35 \sqrt{374946}}{2487} - \frac{375}{829}$$
y2 = -35*sqrt(374946)/2487 - 375/829
Sum and product of roots
[src]
sum
________ ________ 375 35*\/ 374946 375 35*\/ 374946 - --- + ------------- + - --- - ------------- 829 2487 829 2487
$$\left(- \frac{35 \sqrt{374946}}{2487} - \frac{375}{829}\right) + \left(- \frac{375}{829} + \frac{35 \sqrt{374946}}{2487}\right)$$
=
-750 ----- 829
$$- \frac{750}{829}$$
product
/ ________\ / ________\ | 375 35*\/ 374946 | | 375 35*\/ 374946 | |- --- + -------------|*|- --- - -------------| \ 829 2487 / \ 829 2487 /
$$\left(- \frac{375}{829} + \frac{35 \sqrt{374946}}{2487}\right) \left(- \frac{35 \sqrt{374946}}{2487} - \frac{375}{829}\right)$$
=
-184175 -------- 2487
$$- \frac{184175}{2487}$$
-184175/2487