Solve 2z^3 = 5z - 3

Question

Solve the equation

z_{1} = - \frac{1 + 7}{2}, z_{2} = \frac{- 1 + 7}{2}, z_{3} = 1

Alternative Form

z_{1} \approx - 1.822876, z_{2} \approx 0.822876, z_{3} = 1

Evaluate

2 z^{3} = 5 z - 3

Move the expression to the left side

2 z^{3} - (5 z - 3) = 0

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

2 z^{3} - 5 z + 3 = 0

Factor the expression

(z - 1) (2 z^{2} + 2 z - 3) = 0

Separate the equation into 2 possible cases

z - 1 = 0 2 z^{2} + 2 z - 3 = 0

Solve the equation

More Steps

Evaluate

z - 1 = 0

Move the constant to the right-hand side and change its sign

z = 0 + 1

Removing 0 doesn't change the value,so remove it from the expression

z = 1

z = 1 2 z^{2} + 2 z - 3 = 0

Solve the equation

More Steps

Evaluate

2 z^{2} + 2 z - 3 = 0

Substitute a= 2,b= 2 and c= - 3 into the quadratic formula z = \frac{- b \pm b ^{2} - 4 a c}{2 a}

z = \frac{- 2 \pm 2 ^{2} - 4 \times 2 ( - 3 )}{2 \times 2}

Simplify the expression

z = \frac{- 2 \pm 2 ^{2} - 4 \times 2 ( - 3 )}{4}

Simplify the expression

More Steps

Evaluate

2^{2} - 4 \times 2 (- 3)

Multiply

2^{2} - (- 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

2^{2} + 24

Evaluate the power

4 + 24

Add the numbers

28

z = \frac{- 2 \pm 28}{4}

Simplify the radical expression

More Steps

Evaluate

28

Write the expression as a product where the root of one of the factors can be evaluated

4 \times 7

Write the number in exponential form with the base of 2

2^{2} \times 7

The root of a product is equal to the product of the roots of each factor

2^{2} \times 7

Reduce the index of the radical and exponent with 2

27

z = \frac{- 2 \pm 2 7}{4}

Separate the equation into 2 possible cases

z = \frac{- 2 + 2 7}{4} z = \frac{- 2 - 2 7}{4}

Simplify the expression

z = \frac{- 1 + 7}{2} z = \frac{- 2 - 2 7}{4}

Simplify the expression

z = \frac{- 1 + 7}{2} z = - \frac{1 + 7}{2}

z = 1 z = \frac{- 1 + 7}{2} z = - \frac{1 + 7}{2}

Solution

z_{1} = - \frac{1 + 7}{2}, z_{2} = \frac{- 1 + 7}{2}, z_{3} = 1

Alternative Form

z_{1} \approx - 1.822876, z_{2} \approx 0.822876, z_{3} = 1

Show Solution

Solve the equation

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