# Solve for x (2x-1)dx+(3y+7)dy=0 (2x-1)dx+(3y+7)dy=0
Simplify each term.
Apply the distributive property.
(2xd-1d)x+(3y+7)dy=0
Rewrite -1d as -d.
(2xd-d)x+(3y+7)dy=0
Apply the distributive property.
2xdx-dx+(3y+7)dy=0
Multiply x by x by adding the exponents.
Move x.
2(x⋅x)d-dx+(3y+7)dy=0
Multiply x by x.
2x2d-dx+(3y+7)dy=0
2x2d-dx+(3y+7)dy=0
Apply the distributive property.
2x2d-dx+(3yd+7d)y=0
Apply the distributive property.
2x2d-dx+3ydy+7dy=0
Multiply y by y by adding the exponents.
Move y.
2x2d-dx+3(y⋅y)d+7dy=0
Multiply y by y.
2x2d-dx+3y2d+7dy=0
2x2d-dx+3y2d+7dy=0
2x2d-dx+3y2d+7dy=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=2d, b=-d, and c=3y2d+7dy into the quadratic formula and solve for x.
d±(-d)2-4⋅(2d⋅(3y2d+7dy))2(2d)
Simplify.
Simplify the numerator.
Apply the product rule to -d.
x=d±(-1)2d2-4⋅(2d⋅(3y2d+7dy))2⋅(2d)
Raise -1 to the power of 2.
x=d±1d2-4⋅(2d⋅(3y2d+7dy))2⋅(2d)
Multiply d2 by 1.
x=d±d2-4⋅(2d⋅(3y2d+7dy))2⋅(2d)
Apply the distributive property.
x=d±d2-4⋅(2d(3y2d)+2d(7dy))2⋅(2d)
Multiply d by d by adding the exponents.
Move d.
x=d±d2-4⋅(2(d⋅d)(3y2)+2d(7dy))2⋅(2d)
Multiply d by d.
x=d±d2-4⋅(2d2(3y2)+2d(7dy))2⋅(2d)
x=d±d2-4⋅(2d2(3y2)+2d(7dy))2⋅(2d)
Multiply d by d by adding the exponents.
Move d.
x=d±d2-4⋅(2d2(3y2)+2(d⋅d)(7y))2⋅(2d)
Multiply d by d.
x=d±d2-4⋅(2d2(3y2)+2d2(7y))2⋅(2d)
x=d±d2-4⋅(2d2(3y2)+2d2(7y))2⋅(2d)
Simplify each term.
Rewrite using the commutative property of multiplication.
x=d±d2-4⋅(2⋅(3(d2y2))+2d2(7y))2⋅(2d)
Multiply 2 by 3.
x=d±d2-4⋅(6(d2y2)+2d2(7y))2⋅(2d)
Rewrite using the commutative property of multiplication.
x=d±d2-4⋅(6d2y2+2⋅(7(d2y)))2⋅(2d)
Multiply 2 by 7.
x=d±d2-4⋅(6d2y2+14d2y)2⋅(2d)
x=d±d2-4⋅(6d2y2+14d2y)2⋅(2d)
Apply the distributive property.
x=d±d2-4(6d2y2)-4(14d2y)2⋅(2d)
Multiply 6 by -4.
x=d±d2-24(d2y2)-4(14d2y)2⋅(2d)
Multiply 14 by -4.
x=d±d2-24(d2y2)-56(d2y)2⋅(2d)
Remove parentheses.
x=d±d2-24d2y2-56d2y2⋅(2d)
Rewrite d2-24d2y2-56d2y in a factored form.
Factor d2 out of d2-24d2y2-56d2y.
Multiply by 1.
x=d±d2⋅1-24d2y2-56d2y2⋅(2d)
Factor d2 out of -24d2y2.
x=d±d2⋅1+d2(-24y2)-56d2y2⋅(2d)
Factor d2 out of -56d2y.
x=d±d2⋅1+d2(-24y2)+d2(-56y)2⋅(2d)
Factor d2 out of d2⋅1+d2(-24y2).
x=d±d2⋅(1-24y2)+d2(-56y)2⋅(2d)
Factor d2 out of d2⋅(1-24y2)+d2(-56y).
x=d±d2(1-24y2-56y)2⋅(2d)
x=d±d2(1-24y2-56y)2⋅(2d)
Reorder terms.
x=d±d2(-24y2-56y+1)2⋅(2d)
x=d±d2(-24y2-56y+1)2⋅(2d)
Rewrite 1 as 12.
x=d±d2(-24y2-56y+12)2⋅(2d)
Pull terms out from under the radical.
x=d±d-24y2-56y+122⋅(2d)
One to any power is one.
x=d±d-24y2-56y+12⋅(2d)
x=d±d-24y2-56y+12⋅(2d)
Multiply 2 by 2.
x=d±d-24y2-56y+14d
Simplify d±d-24y2-56y+14d.
x=1±-24y2-56y+14
x=1±-24y2-56y+14
The final answer is the combination of both solutions.
x=1+-24y2-56y+14
x=1–24y2-56y+14
Solve for x (2x-1)dx+(3y+7)dy=0

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