Solve 1/2 *(3-c)^2(3c-5)-10c(3-c)

Question

Simplify the expression

- \frac{3}{2} c - \frac{3}{2} c^{2} + \frac{3}{2} c^{3} - \frac{45}{2}

Evaluate

\frac{1}{2} (3 - c)^{2} (3 c - 5) - 10 c (3 - c)

Multiply the terms

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Multiply the terms

\frac{1}{2} (3 - c)^{2} (3 c - 5)

Multiply the terms

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Evaluate

\frac{1}{2} (3 c - 5)

Apply the distributive property

\frac{1}{2} \times 3 c - \frac{1}{2} \times 5

Multiply the numbers

\frac{3}{2} c - \frac{1}{2} \times 5

Multiply the numbers

\frac{3}{2} c - \frac{5}{2}

(\frac{3}{2} c - \frac{5}{2}) (3 - c)^{2}

(\frac{3}{2} c - \frac{5}{2}) (3 - c)^{2} - 10 c (3 - c)

Expand the expression

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Calculate

(\frac{3}{2} c - \frac{5}{2}) (3 - c)^{2}

Simplify

(\frac{3}{2} c - \frac{5}{2}) (9 - 6 c + c^{2})

Apply the distributive property

\frac{3}{2} c \times 9 - \frac{3}{2} c \times 6 c + \frac{3}{2} c \times c^{2} - \frac{5}{2} \times 9 - (- \frac{5}{2} \times 6 c) - \frac{5}{2} c^{2}

Multiply the numbers

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Evaluate

\frac{3}{2} \times 9

Multiply the numbers

\frac{3 \times 9}{2}

Multiply the numbers

\frac{27}{2}

\frac{27}{2} c - \frac{3}{2} c \times 6 c + \frac{3}{2} c \times c^{2} - \frac{5}{2} \times 9 - (- \frac{5}{2} \times 6 c) - \frac{5}{2} c^{2}

Multiply the terms

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Evaluate

\frac{3}{2} c \times 6 c

Multiply the numbers

9 c \times c

Multiply the terms

9 c^{2}

\frac{27}{2} c - 9 c^{2} + \frac{3}{2} c \times c^{2} - \frac{5}{2} \times 9 - (- \frac{5}{2} \times 6 c) - \frac{5}{2} c^{2}

Multiply the terms

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Evaluate

c \times c^{2}

Use the product rule a^{n} \times a^{m} = a^{n + m} to simplify the expression

c^{1 + 2}

Add the numbers

c^{3}

\frac{27}{2} c - 9 c^{2} + \frac{3}{2} c^{3} - \frac{5}{2} \times 9 - (- \frac{5}{2} \times 6 c) - \frac{5}{2} c^{2}

Multiply the numbers

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Evaluate

- \frac{5}{2} \times 9

Multiply the numbers

- \frac{5 \times 9}{2}

Multiply the numbers

- \frac{45}{2}

\frac{27}{2} c - 9 c^{2} + \frac{3}{2} c^{3} - \frac{45}{2} - (- \frac{5}{2} \times 6 c) - \frac{5}{2} c^{2}

Multiply the numbers

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Evaluate

- \frac{5}{2} \times 6

Reduce the numbers

- 5 \times 3

Multiply the numbers

- 15

\frac{27}{2} c - 9 c^{2} + \frac{3}{2} c^{3} - \frac{45}{2} - (- 15 c) - \frac{5}{2} c^{2}

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

\frac{27}{2} c - 9 c^{2} + \frac{3}{2} c^{3} - \frac{45}{2} + 15 c - \frac{5}{2} c^{2}

Add the terms

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Evaluate

\frac{27}{2} c + 15 c

Collect like terms by calculating the sum or difference of their coefficients

(\frac{27}{2} + 15) c

Add the numbers

\frac{57}{2} c

\frac{57}{2} c - 9 c^{2} + \frac{3}{2} c^{3} - \frac{45}{2} - \frac{5}{2} c^{2}

Subtract the terms

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Evaluate

- 9 c^{2} - \frac{5}{2} c^{2}

Collect like terms by calculating the sum or difference of their coefficients

(- 9 - \frac{5}{2}) c^{2}

Subtract the numbers

- \frac{23}{2} c^{2}

\frac{57}{2} c - \frac{23}{2} c^{2} + \frac{3}{2} c^{3} - \frac{45}{2}

\frac{57}{2} c - \frac{23}{2} c^{2} + \frac{3}{2} c^{3} - \frac{45}{2} - 10 c (3 - c)

Expand the expression

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Calculate

- 10 c (3 - c)

Apply the distributive property

- 10 c \times 3 - (- 10 c \times c)

Multiply the numbers

- 30 c - (- 10 c \times c)

Multiply the terms

- 30 c - (- 10 c^{2})

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

- 30 c + 10 c^{2}

\frac{57}{2} c - \frac{23}{2} c^{2} + \frac{3}{2} c^{3} - \frac{45}{2} - 30 c + 10 c^{2}

Subtract the terms

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Evaluate

\frac{57}{2} c - 30 c

Collect like terms by calculating the sum or difference of their coefficients

(\frac{57}{2} - 30) c

Subtract the numbers

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Evaluate

\frac{57}{2} - 30

Reduce fractions to a common denominator

\frac{57}{2} - \frac{30 \times 2}{2}

Write all numerators above the common denominator

\frac{57 - 30 \times 2}{2}

Multiply the numbers

\frac{57 - 60}{2}

Subtract the numbers

\frac{- 3}{2}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{3}{2}

- \frac{3}{2} c

- \frac{3}{2} c - \frac{23}{2} c^{2} + \frac{3}{2} c^{3} - \frac{45}{2} + 10 c^{2}

Solution

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Evaluate

- \frac{23}{2} c^{2} + 10 c^{2}

Collect like terms by calculating the sum or difference of their coefficients

(- \frac{23}{2} + 10) c^{2}

Add the numbers

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Evaluate

- \frac{23}{2} + 10

Reduce fractions to a common denominator

- \frac{23}{2} + \frac{10 \times 2}{2}

Write all numerators above the common denominator

\frac{- 23 + 10 \times 2}{2}

Multiply the numbers

\frac{- 23 + 20}{2}

Add the numbers

\frac{- 3}{2}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{3}{2}

- \frac{3}{2} c^{2}

- \frac{3}{2} c - \frac{3}{2} c^{2} + \frac{3}{2} c^{3} - \frac{45}{2}

Show Solution

Factor the expression

- \frac{3}{2} (2 c + 5 + c^{2}) (3 - c)

Evaluate

\frac{1}{2} (3 - c)^{2} (3 c - 5) - 10 c (3 - c)

Rewrite the expression

\frac{1}{2} (3 - c) (3 c - 5) (3 - c) - 10 c (3 - c)

Factor out 3 - c from the expression

(\frac{1}{2} (3 - c) (3 c - 5) - 10 c) (3 - c)

Solution

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Evaluate

\frac{1}{2} (3 - c) (3 c - 5) - 10 c

Simplify

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Evaluate

\frac{1}{2} (3 - c) (3 c - 5)

Simplify

(\frac{3}{2} - \frac{1}{2} c) (3 c - 5)

Apply the distributive property

\frac{3}{2} \times 3 c + \frac{3}{2} (- 5) - \frac{1}{2} c \times 3 c - \frac{1}{2} c (- 5)

Multiply the terms

\frac{9}{2} c + \frac{3}{2} (- 5) - \frac{1}{2} c \times 3 c - \frac{1}{2} c (- 5)

Multiply the terms

\frac{9}{2} c - \frac{15}{2} - \frac{1}{2} c \times 3 c - \frac{1}{2} c (- 5)

Multiply the terms

\frac{9}{2} c - \frac{15}{2} - \frac{3}{2} c^{2} - \frac{1}{2} c (- 5)

Multiply the terms

\frac{9}{2} c - \frac{15}{2} - \frac{3}{2} c^{2} + \frac{5}{2} c

\frac{9}{2} c - \frac{15}{2} - \frac{3}{2} c^{2} + \frac{5}{2} c - 10 c

Calculate the sum or difference

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Evaluate

\frac{9}{2} c + \frac{5}{2} c - 10 c

Collect like terms by calculating the sum or difference of their coefficients

(\frac{9}{2} + \frac{5}{2} - 10) c

Calculate the sum or difference

- 3 c

- 3 c - \frac{15}{2} - \frac{3}{2} c^{2}

Factor the expression

- \frac{3}{2} (2 c + 5 + c^{2})

- \frac{3}{2} (2 c + 5 + c^{2}) (3 - c)

Show Solution

Find the roots

c_{1} = - 1 - 2 i, c_{2} = - 1 + 2 i, c_{3} = 3

Evaluate

\frac{1}{2} (3 - c)^{2} (3 c - 5) - 10 c (3 - c)

To find the roots of the expression,set the expression equal to 0

\frac{1}{2} (3 - c)^{2} (3 c - 5) - 10 c (3 - c) = 0

Multiply the terms

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Multiply the terms

\frac{1}{2} (3 - c)^{2} (3 c - 5)

Multiply the terms

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Evaluate

\frac{1}{2} (3 c - 5)

Apply the distributive property

\frac{1}{2} \times 3 c - \frac{1}{2} \times 5

Multiply the numbers

\frac{3}{2} c - \frac{1}{2} \times 5

Multiply the numbers

\frac{3}{2} c - \frac{5}{2}

(\frac{3}{2} c - \frac{5}{2}) (3 - c)^{2}

(\frac{3}{2} c - \frac{5}{2}) (3 - c)^{2} - 10 c (3 - c) = 0

Calculate

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Evaluate

(\frac{3}{2} c - \frac{5}{2}) (3 - c)^{2} - 10 c (3 - c)

Expand the expression

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Calculate

(\frac{3}{2} c - \frac{5}{2}) (3 - c)^{2}

Simplify

(\frac{3}{2} c - \frac{5}{2}) (9 - 6 c + c^{2})

Apply the distributive property

\frac{3}{2} c \times 9 - \frac{3}{2} c \times 6 c + \frac{3}{2} c \times c^{2} - \frac{5}{2} \times 9 - (- \frac{5}{2} \times 6 c) - \frac{5}{2} c^{2}

Multiply the numbers

\frac{27}{2} c - \frac{3}{2} c \times 6 c + \frac{3}{2} c \times c^{2} - \frac{5}{2} \times 9 - (- \frac{5}{2} \times 6 c) - \frac{5}{2} c^{2}

Multiply the terms

\frac{27}{2} c - 9 c^{2} + \frac{3}{2} c \times c^{2} - \frac{5}{2} \times 9 - (- \frac{5}{2} \times 6 c) - \frac{5}{2} c^{2}

Multiply the terms

\frac{27}{2} c - 9 c^{2} + \frac{3}{2} c^{3} - \frac{5}{2} \times 9 - (- \frac{5}{2} \times 6 c) - \frac{5}{2} c^{2}

Multiply the numbers

\frac{27}{2} c - 9 c^{2} + \frac{3}{2} c^{3} - \frac{45}{2} - (- \frac{5}{2} \times 6 c) - \frac{5}{2} c^{2}

Multiply the numbers

\frac{27}{2} c - 9 c^{2} + \frac{3}{2} c^{3} - \frac{45}{2} - (- 15 c) - \frac{5}{2} c^{2}

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

\frac{27}{2} c - 9 c^{2} + \frac{3}{2} c^{3} - \frac{45}{2} + 15 c - \frac{5}{2} c^{2}

Add the terms

\frac{57}{2} c - 9 c^{2} + \frac{3}{2} c^{3} - \frac{45}{2} - \frac{5}{2} c^{2}

Subtract the terms

\frac{57}{2} c - \frac{23}{2} c^{2} + \frac{3}{2} c^{3} - \frac{45}{2}

\frac{57}{2} c - \frac{23}{2} c^{2} + \frac{3}{2} c^{3} - \frac{45}{2} - 10 c (3 - c)

Expand the expression

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Calculate

- 10 c (3 - c)

Apply the distributive property

- 10 c \times 3 - (- 10 c \times c)

Multiply the numbers

- 30 c - (- 10 c \times c)

Multiply the terms

- 30 c - (- 10 c^{2})

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

- 30 c + 10 c^{2}

\frac{57}{2} c - \frac{23}{2} c^{2} + \frac{3}{2} c^{3} - \frac{45}{2} - 30 c + 10 c^{2}

Subtract the terms

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Evaluate

\frac{57}{2} c - 30 c

Collect like terms by calculating the sum or difference of their coefficients

(\frac{57}{2} - 30) c

Subtract the numbers

- \frac{3}{2} c

- \frac{3}{2} c - \frac{23}{2} c^{2} + \frac{3}{2} c^{3} - \frac{45}{2} + 10 c^{2}

Add the terms

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Evaluate

- \frac{23}{2} c^{2} + 10 c^{2}

Collect like terms by calculating the sum or difference of their coefficients

(- \frac{23}{2} + 10) c^{2}

Add the numbers

- \frac{3}{2} c^{2}

- \frac{3}{2} c - \frac{3}{2} c^{2} + \frac{3}{2} c^{3} - \frac{45}{2}

- \frac{3}{2} c - \frac{3}{2} c^{2} + \frac{3}{2} c^{3} - \frac{45}{2} = 0

Factor the expression

\frac{3}{2} (c - 3) (2 c + c^{2} + 5) = 0

Divide both sides

(c - 3) (2 c + c^{2} + 5) = 0

Separate the equation into 2 possible cases

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

Solve the equation

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Evaluate

c - 3 = 0

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

c = 0 + 3

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

c = 3

c = 3 2 c + c^{2} + 5 = 0

Solve the equation

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Evaluate

2 c + c^{2} + 5 = 0

Rewrite in standard form

c^{2} + 2 c + 5 = 0

Substitute a= 1,b= 2 and c= 5 into the quadratic formula c = \frac{- b \pm b ^{2} - 4 a c}{2 a}

c = \frac{- 2 \pm 2 ^{2} - 4 \times 5}{2}

Simplify the expression

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Evaluate

2^{2} - 4 \times 5

Multiply the numbers

2^{2} - 20

Evaluate the power

4 - 20

Subtract the numbers

- 16

c = \frac{- 2 \pm - 16}{2}

Simplify the radical expression

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Evaluate

- 16

Evaluate the power

16 \times - 1

Evaluate the power

16 \times i

Evaluate the square root

4 i

c = \frac{- 2 \pm 4 i}{2}

Separate the equation into 2 possible cases

c = \frac{- 2 + 4 i}{2} c = \frac{- 2 - 4 i}{2}

Simplify the expression

c = - 1 + 2 i c = \frac{- 2 - 4 i}{2}

Simplify the expression

c = - 1 + 2 i c = - 1 - 2 i

c = 3 c = - 1 + 2 i c = - 1 - 2 i

Solution

c_{1} = - 1 - 2 i, c_{2} = - 1 + 2 i, c_{3} = 3

Show Solution