Solve for z 68=140z-16z^2

Math
68=140z-16z2
Rewrite the equation as 140z-16z2=68.
140z-16z2=68
Move 68 to the left side of the equation by subtracting it from both sides.
140z-16z2-68=0
Factor the left side of the equation.
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Factor 4 out of 140z-16z2-68.
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Factor 4 out of 140z.
4(35z)-16z2-68
Factor 4 out of -16z2.
4(35z)+4(-4z2)-68
Factor 4 out of -68.
4(35z)+4(-4z2)+4(-17)
Factor 4 out of 4(35z)+4(-4z2).
4(35z-4z2)+4(-17)
Factor 4 out of 4(35z-4z2)+4(-17).
4(35z-4z2-17)
4(35z-4z2-17)
Reorder terms.
4(-4z2+35z-17)
Replace the left side with the factored expression.
4(-4z2+35z-17)=0
4(-4z2+35z-17)=0
Divide each term by 4 and simplify.
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Divide each term in 4(-4z2+35z-17)=0 by 4.
4(-4z2+35z-17)4=04
Cancel the common factor of 4.
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Cancel the common factor.
4(-4z2+35z-17)4=04
Divide -4z2+35z-17 by 1.
-4z2+35z-17=04
-4z2+35z-17=04
Divide 0 by 4.
-4z2+35z-17=0
-4z2+35z-17=0
Factor -1 out of -4z2+35z-17.
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Factor -1 out of -4z2.
-(4z2)+35z-17=0
Factor -1 out of 35z.
-(4z2)-(-35z)-17=0
Rewrite -17 as -1(17).
-(4z2)-(-35z)-1⋅17=0
Factor -1 out of -(4z2)-(-35z).
-(4z2-35z)-1⋅17=0
Factor -1 out of -(4z2-35z)-1(17).
-(4z2-35z+17)=0
-(4z2-35z+17)=0
Multiply each term in -(4z2-35z+17)=0 by -1
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Multiply each term in -(4z2-35z+17)=0 by -1.
-(4z2-35z+17)⋅-1=0⋅-1
Simplify -(4z2-35z+17)⋅-1.
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Apply the distributive property.
(-(4z2)-(-35z)-1⋅17)⋅-1=0⋅-1
Simplify.
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Multiply 4 by -1.
(-4z2-(-35z)-1⋅17)⋅-1=0⋅-1
Multiply -35 by -1.
(-4z2+35z-1⋅17)⋅-1=0⋅-1
Multiply -1 by 17.
(-4z2+35z-17)⋅-1=0⋅-1
(-4z2+35z-17)⋅-1=0⋅-1
Apply the distributive property.
-4z2⋅-1+35z⋅-1-17⋅-1=0⋅-1
Simplify.
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Multiply -1 by -4.
4z2+35z⋅-1-17⋅-1=0⋅-1
Multiply -1 by 35.
4z2-35z-17⋅-1=0⋅-1
Multiply -17 by -1.
4z2-35z+17=0⋅-1
4z2-35z+17=0⋅-1
4z2-35z+17=0⋅-1
Multiply 0 by -1.
4z2-35z+17=0
4z2-35z+17=0
Use the quadratic formula to find the solutions.
-b±b2-4(ac)2a
Substitute the values a=4, b=-35, and c=17 into the quadratic formula and solve for z.
35±(-35)2-4⋅(4⋅17)2⋅4
Simplify.
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Simplify the numerator.
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Raise -35 to the power of 2.
z=35±1225-4⋅(4⋅17)2⋅4
Multiply 4 by 17.
z=35±1225-4⋅682⋅4
Multiply -4 by 68.
z=35±1225-2722⋅4
Subtract 272 from 1225.
z=35±9532⋅4
z=35±9532⋅4
Multiply 2 by 4.
z=35±9538
z=35±9538
The final answer is the combination of both solutions.
z=35+9538,35-9538
The result can be shown in multiple forms.
Exact Form:
z=35+9538,35-9538
Decimal Form:
z=8.23383726…,0.51616273…
Solve for z 68=140z-16z^2

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