Algebra
Calculus
Trigonometry
Matrix
Question
Solve the equation
Solve for x
Solve for y
Solve for z
x=2+−2−y2+6y−z2−2zx=2−−2−y2+6y−z2−2z
Evaluate
x2+y2+z2−4x−6y+2z+6=0
Rewrite the expression
x2+y2+z2−6y+2z+6−4x=0
Rewrite in standard form
x2−4x+y2+z2−6y+2z+6=0
Substitute a=1,b=−4 and c=y2+z2−6y+2z+6 into the quadratic formula x=2a−b±b2−4ac
x=24±(−4)2−4(y2+z2−6y+2z+6)
Simplify the expression
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Evaluate
(−4)2−4(y2+z2−6y+2z+6)
Multiply the terms
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Evaluate
4(y2+z2−6y+2z+6)
Apply the distributive property
4y2+4z2−4×6y+4×2z+4×6
Multiply the terms
4y2+4z2−24y+4×2z+4×6
Multiply the terms
4y2+4z2−24y+8z+4×6
Multiply the numbers
4y2+4z2−24y+8z+24
(−4)2−(4y2+4z2−24y+8z+24)
Rewrite the expression
42−(4y2+4z2−24y+8z+24)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
42−4y2−4z2+24y−8z−24
Evaluate the power
16−4y2−4z2+24y−8z−24
Subtract the numbers
−8−4y2−4z2+24y−8z
x=24±−8−4y2−4z2+24y−8z
Simplify the radical expression
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Evaluate
−8−4y2−4z2+24y−8z
Factor the expression
4(−2−y2−z2+6y−2z)
The root of a product is equal to the product of the roots of each factor
4×−2−y2−z2+6y−2z
Evaluate the root
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Evaluate
4
Write the number in exponential form with the base of 2
22
Reduce the index of the radical and exponent with 2
2
2−2−y2−z2+6y−2z
Simplify
2−2−y2+6y−z2−2z
x=24±2−2−y2+6y−z2−2z
Separate the equation into 2 possible cases
x=24+2−2−y2+6y−z2−2zx=24−2−2−y2+6y−z2−2z
Simplify the expression
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Evaluate
x=24+2−2−y2+6y−z2−2z
Divide the terms
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Evaluate
24+2−2−y2+6y−z2−2z
Rewrite the expression
22(2+−2−y2+6y−z2−2z)
Reduce the fraction
2+−2−y2+6y−z2−2z
x=2+−2−y2+6y−z2−2z
x=2+−2−y2+6y−z2−2zx=24−2−2−y2+6y−z2−2z
Solution
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Evaluate
x=24−2−2−y2+6y−z2−2z
Divide the terms
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Evaluate
24−2−2−y2+6y−z2−2z
Rewrite the expression
22(2−−2−y2+6y−z2−2z)
Reduce the fraction
2−−2−y2+6y−z2−2z
x=2−−2−y2+6y−z2−2z
x=2+−2−y2+6y−z2−2zx=2−−2−y2+6y−z2−2z
Show Solution
Find the partial derivative
Find ∂x∂z by differentiating the equation directly
Find ∂y∂z by differentiating the equation directly
∂x∂z=z+1−x+2
Evaluate
x2+y2+z2−4x−6y+2z+6=0
Find ∂x∂z by taking the derivative of both sides with respect to x
∂x∂(x2+y2+z2−4x−6y+2z+6)=∂x∂(0)
Use differentiation rule ∂x∂(f(x)±g(x))=∂x∂(f(x))±∂x∂(g(x))
∂x∂(x2)+∂x∂(y2)+∂x∂(z2)−∂x∂(4x)−∂x∂(6y)+∂x∂(2z)+∂x∂(6)=∂x∂(0)
Use ∂x∂xn=nxn−1 to find derivative
2x+∂x∂(y2)+∂x∂(z2)−∂x∂(4x)−∂x∂(6y)+∂x∂(2z)+∂x∂(6)=∂x∂(0)
Use ∂x∂(c)=0 to find derivative
2x+0+∂x∂(z2)−∂x∂(4x)−∂x∂(6y)+∂x∂(2z)+∂x∂(6)=∂x∂(0)
Evaluate
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Evaluate
∂x∂(z2)
Use the chain rule ∂x∂(f(g))=∂g∂(f(g))×∂x∂(g) where the g=z, to find the derivative
∂z∂(z2)∂x∂z
Find the derivative
2z∂x∂z
2x+0+2z∂x∂z−∂x∂(4x)−∂x∂(6y)+∂x∂(2z)+∂x∂(6)=∂x∂(0)
Evaluate
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Evaluate
∂x∂(4x)
Use differentiation rule ∂x∂(cf(x))=c×∂x∂(f(x))
4×∂x∂(x)
Use ∂x∂xn=nxn−1 to find derivative
4×1
Multiply the terms
4
2x+0+2z∂x∂z−4−∂x∂(6y)+∂x∂(2z)+∂x∂(6)=∂x∂(0)
Use ∂x∂(c)=0 to find derivative
2x+0+2z∂x∂z−4−0+∂x∂(2z)+∂x∂(6)=∂x∂(0)
Evaluate
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Evaluate
∂x∂(2z)
Use differentiation rule ∂x∂(cf(x))=c×∂x∂(f(x))
2×∂x∂(z)
Find the derivative
2∂x∂z
2x+0+2z∂x∂z−4−0+2∂x∂z+∂x∂(6)=∂x∂(0)
Use ∂x∂(c)=0 to find derivative
2x+0+2z∂x∂z−4−0+2∂x∂z+0=∂x∂(0)
Removing 0 doesn't change the value,so remove it from the expression
2x+2z∂x∂z−4+2∂x∂z=∂x∂(0)
Find the partial derivative
2x+2z∂x∂z−4+2∂x∂z=0
Rewrite the expression
2x−4+2z∂x∂z+2∂x∂z=0
Collect like terms by calculating the sum or difference of their coefficients
2x−4+(2z+2)∂x∂z=0
Move the constant to the right side
(2z+2)∂x∂z=0−(2x−4)
Subtract the terms
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Evaluate
0−(2x−4)
Removing 0 doesn't change the value,so remove it from the expression
−(2x−4)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
−2x+4
(2z+2)∂x∂z=−2x+4
Divide both sides
2z+2(2z+2)∂x∂z=2z+2−2x+4
Divide the numbers
∂x∂z=2z+2−2x+4
Solution
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Evaluate
2z+2−2x+4
Rewrite the expression
2z+22(−x+2)
Rewrite the expression
2(z+1)2(−x+2)
Reduce the fraction
z+1−x+2
∂x∂z=z+1−x+2
Show Solution