Solve (3k-2)(3k^2-3k-3)

Question

Simplify the expression

9 k^{3} - 15 k^{2} - 3 k + 6

Evaluate

(3 k - 2) (3 k^{2} - 3 k - 3)

Apply the distributive property

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

Multiply the terms

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Evaluate

3 k \times 3 k^{2}

Multiply the numbers

9 k \times k^{2}

Multiply the terms

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Evaluate

k \times k^{2}

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

k^{1 + 2}

Add the numbers

k^{3}

9 k^{3}

9 k^{3} - 3 k \times 3 k - 3 k \times 3 - 2 \times 3 k^{2} - (- 2 \times 3 k) - (- 2 \times 3)

Multiply the terms

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Evaluate

3 k \times 3 k

Multiply the numbers

9 k \times k

Multiply the terms

9 k^{2}

9 k^{3} - 9 k^{2} - 3 k \times 3 - 2 \times 3 k^{2} - (- 2 \times 3 k) - (- 2 \times 3)

Multiply the numbers

9 k^{3} - 9 k^{2} - 9 k - 2 \times 3 k^{2} - (- 2 \times 3 k) - (- 2 \times 3)

Multiply the numbers

9 k^{3} - 9 k^{2} - 9 k - 6 k^{2} - (- 2 \times 3 k) - (- 2 \times 3)

Multiply the numbers

9 k^{3} - 9 k^{2} - 9 k - 6 k^{2} - (- 6 k) - (- 2 \times 3)

Multiply the numbers

9 k^{3} - 9 k^{2} - 9 k - 6 k^{2} - (- 6 k) - (- 6)

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

9 k^{3} - 9 k^{2} - 9 k - 6 k^{2} + 6 k + 6

Subtract the terms

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Evaluate

- 9 k^{2} - 6 k^{2}

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

(- 9 - 6) k^{2}

Subtract the numbers

- 15 k^{2}

9 k^{3} - 15 k^{2} - 9 k + 6 k + 6

Solution

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Evaluate

- 9 k + 6 k

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

(- 9 + 6) k

Add the numbers

- 3 k

9 k^{3} - 15 k^{2} - 3 k + 6

Show Solution

Factor the expression

3 (3 k - 2) (k^{2} - k - 1)

Evaluate

(3 k - 2) (3 k^{2} - 3 k - 3)

Factor the expression

(3 k - 2) \times 3 (k^{2} - k - 1)

Solution

3 (3 k - 2) (k^{2} - k - 1)

Show Solution

Find the roots

k_{1} = \frac{1 - 5}{2}, k_{2} = \frac{2}{3}, k_{3} = \frac{1 + 5}{2}

Alternative Form

k_{1} \approx - 0.618034, k_{2} = 0. \dot{6}, k_{3} \approx 1.618034

Evaluate

(3 k - 2) (3 k^{2} - 3 k - 3)

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

(3 k - 2) (3 k^{2} - 3 k - 3) = 0

Separate the equation into 2 possible cases

3 k - 2 = 0 3 k^{2} - 3 k - 3 = 0

Solve the equation

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Evaluate

3 k - 2 = 0

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

3 k = 0 + 2

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

3 k = 2

Divide both sides

\frac{3 k}{3} = \frac{2}{3}

Divide the numbers

k = \frac{2}{3}

k = \frac{2}{3} 3 k^{2} - 3 k - 3 = 0

Solve the equation

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Evaluate

3 k^{2} - 3 k - 3 = 0

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

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

Simplify the expression

k = \frac{3 \pm ( - 3 ) ^{2} - 4 \times 3 ( - 3 )}{6}

Simplify the expression

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Evaluate

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

Multiply

(- 3)^{2} - (- 36)

Rewrite the expression

3^{2} - (- 36)

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

3^{2} + 36

Evaluate the power

9 + 36

Add the numbers

45

k = \frac{3 \pm 45}{6}

Simplify the radical expression

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Evaluate

45

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

9 \times 5

Write the number in exponential form with the base of 3

3^{2} \times 5

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

3^{2} \times 5

Reduce the index of the radical and exponent with 2

35

k = \frac{3 \pm 3 5}{6}

Separate the equation into 2 possible cases

k = \frac{3 + 3 5}{6} k = \frac{3 - 3 5}{6}

Simplify the expression

k = \frac{1 + 5}{2} k = \frac{3 - 3 5}{6}

Simplify the expression

k = \frac{1 + 5}{2} k = \frac{1 - 5}{2}

k = \frac{2}{3} k = \frac{1 + 5}{2} k = \frac{1 - 5}{2}

Solution

k_{1} = \frac{1 - 5}{2}, k_{2} = \frac{2}{3}, k_{3} = \frac{1 + 5}{2}

Alternative Form

k_{1} \approx - 0.618034, k_{2} = 0. \dot{6}, k_{3} \approx 1.618034

Show Solution