(3z-1)(z-4)=z^2-8z+7

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Solution for (3z-1)(z-4)=z^2-8z+7 equation:


Simplifying
(3z + -1)(z + -4) = z2 + -8z + 7

Reorder the terms:
(-1 + 3z)(z + -4) = z2 + -8z + 7

Reorder the terms:
(-1 + 3z)(-4 + z) = z2 + -8z + 7

Multiply (-1 + 3z) * (-4 + z)
(-1(-4 + z) + 3z * (-4 + z)) = z2 + -8z + 7
((-4 * -1 + z * -1) + 3z * (-4 + z)) = z2 + -8z + 7
((4 + -1z) + 3z * (-4 + z)) = z2 + -8z + 7
(4 + -1z + (-4 * 3z + z * 3z)) = z2 + -8z + 7
(4 + -1z + (-12z + 3z2)) = z2 + -8z + 7

Combine like terms: -1z + -12z = -13z
(4 + -13z + 3z2) = z2 + -8z + 7

Reorder the terms:
4 + -13z + 3z2 = 7 + -8z + z2

Solving
4 + -13z + 3z2 = 7 + -8z + z2

Solving for variable 'z'.

Reorder the terms:
4 + -7 + -13z + 8z + 3z2 + -1z2 = 7 + -8z + z2 + -7 + 8z + -1z2

Combine like terms: 4 + -7 = -3
-3 + -13z + 8z + 3z2 + -1z2 = 7 + -8z + z2 + -7 + 8z + -1z2

Combine like terms: -13z + 8z = -5z
-3 + -5z + 3z2 + -1z2 = 7 + -8z + z2 + -7 + 8z + -1z2

Combine like terms: 3z2 + -1z2 = 2z2
-3 + -5z + 2z2 = 7 + -8z + z2 + -7 + 8z + -1z2

Reorder the terms:
-3 + -5z + 2z2 = 7 + -7 + -8z + 8z + z2 + -1z2

Combine like terms: 7 + -7 = 0
-3 + -5z + 2z2 = 0 + -8z + 8z + z2 + -1z2
-3 + -5z + 2z2 = -8z + 8z + z2 + -1z2

Combine like terms: -8z + 8z = 0
-3 + -5z + 2z2 = 0 + z2 + -1z2
-3 + -5z + 2z2 = z2 + -1z2

Combine like terms: z2 + -1z2 = 0
-3 + -5z + 2z2 = 0

Factor a trinomial.
(-1 + -2z)(3 + -1z) = 0

Subproblem 1

Set the factor '(-1 + -2z)' equal to zero and attempt to solve: Simplifying -1 + -2z = 0 Solving -1 + -2z = 0 Move all terms containing z to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + -2z = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -2z = 0 + 1 -2z = 0 + 1 Combine like terms: 0 + 1 = 1 -2z = 1 Divide each side by '-2'. z = -0.5 Simplifying z = -0.5

Subproblem 2

Set the factor '(3 + -1z)' equal to zero and attempt to solve: Simplifying 3 + -1z = 0 Solving 3 + -1z = 0 Move all terms containing z to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + -1z = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -1z = 0 + -3 -1z = 0 + -3 Combine like terms: 0 + -3 = -3 -1z = -3 Divide each side by '-1'. z = 3 Simplifying z = 3

Solution

z = {-0.5, 3}

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