Solve (x-3)^3-(x-1)^3

Question

Simplify the expression

- 6 x^{2} + 24 x - 26

Evaluate

(x - 3)^{3} - (x - 1)^{3}

Expand the expression

x^{3} - 9 x^{2} + 27 x - 27 - (x - 1)^{3}

Expand the expression

x^{3} - 9 x^{2} + 27 x - 27 - x^{3} + 3 x^{2} - 3 x + 1

The sum of two opposites equals 0

More Steps

Evaluate

x^{3} - x^{3}

Collect like terms

(1 - 1) x^{3}

Add the coefficients

0 \times x^{3}

Calculate

0

0 - 9 x^{2} + 27 x - 27 + 3 x^{2} - 3 x + 1

Remove 0

- 9 x^{2} + 27 x - 27 + 3 x^{2} - 3 x + 1

Add the terms

More Steps

Evaluate

- 9 x^{2} + 3 x^{2}

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

(- 9 + 3) x^{2}

Add the numbers

- 6 x^{2}

- 6 x^{2} + 27 x - 27 - 3 x + 1

Subtract the terms

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Evaluate

27 x - 3 x

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

(27 - 3) x

Subtract the numbers

24 x

- 6 x^{2} + 24 x - 27 + 1

Solution

- 6 x^{2} + 24 x - 26

Show Solution

Factor the expression

- 2 (3 x^{2} - 12 x + 13)

Evaluate

(x - 3)^{3} - (x - 1)^{3}

Use a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2}) to factor the expression

(x - 3 - x + 1) ((x - 3)^{2} + (x - 3) (x - 1) + (x - 1)^{2})

Calculate

More Steps

Simplify

x - 3 - x + 1

The sum of two opposites equals 0

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Evaluate

x - x

Collect like terms

(1 - 1) x

Add the coefficients

0 \times x

Calculate

0

0 - 3 + 1

Remove 0

- 3 + 1

Add the numbers

- 2

- 2 ((x - 3)^{2} + (x - 3) (x - 1) + (x - 1)^{2})

Solution

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Simplify

(x - 3)^{2} + (x - 3) (x - 1) + (x - 1)^{2}

Simplify

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Simplify

(x - 3) (x - 1)

Apply the distributive property

x \times x + x (- 1) - 3 x - 3 (- 1)

Multiply the terms

x^{2} + x (- 1) - 3 x - 3 (- 1)

Multiplying or dividing an odd number of negative terms equals a negative

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

Multiply the terms

x^{2} - x - 3 x + 3

(x - 3)^{2} + x^{2} - x - 3 x + 3 + (x - 1)^{2}

Simplify

More Steps

Evaluate

- x - 3 x

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

(- 1 - 3) x

Subtract the numbers

- 4 x

(x - 3)^{2} + x^{2} - 4 x + 3 + (x - 1)^{2}

Expand the expression

x^{2} - 6 x + 9 + x^{2} - 4 x + 3 + (x - 1)^{2}

Expand the expression

x^{2} - 6 x + 9 + x^{2} - 4 x + 3 + x^{2} - 2 x + 1

Add the terms

More Steps

Evaluate

x^{2} + x^{2} + x^{2}

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

(1 + 1 + 1) x^{2}

Add the numbers

3 x^{2}

3 x^{2} - 6 x + 9 - 4 x + 3 - 2 x + 1

Subtract the terms

More Steps

Evaluate

- 6 x - 4 x - 2 x

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

(- 6 - 4 - 2) x

Subtract the numbers

- 12 x

3 x^{2} - 12 x + 9 + 3 + 1

Add the numbers

More Steps

Evaluate

9 + 3 + 1

Add the numbers

12 + 1

Add the numbers

13

3 x^{2} - 12 x + 13

- 2 (3 x^{2} - 12 x + 13)

Show Solution

Find the roots

x_{1} = 2 - \frac{3}{3} i, x_{2} = 2 + \frac{3}{3} i

Alternative Form

x_{1} \approx 2 - 0.57735 i, x_{2} \approx 2 + 0.57735 i

Evaluate

(x - 3)^{3} - (x - 1)^{3}

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

(x - 3)^{3} - (x - 1)^{3} = 0

Use a^{3} - b^{3} = (a - b) (a^{2} + ab + b^{2}) to factor the expression

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

Separate the equation into 2 possible cases

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

Solve the equation

More Steps

Evaluate

x - 3 - x + 1 = 0

Calculate the sum or difference

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Evaluate

x - 3 - x + 1

The sum of two opposites equals 0

0 - 3 + 1

Remove 0

- 3 + 1

Add the numbers

- 2

- 2 = 0

The statement is false for any value of x

x \in \emptyset

x \in \emptyset (x - 3)^{2} + (x - 3) (x - 1) + (x - 1)^{2} = 0

Solve the equation

More Steps

Evaluate

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

Calculate

More Steps

Evaluate

(x - 3)^{2} + (x - 3) (x - 1) + (x - 1)^{2}

Expand the expression

x^{2} - 6 x + 9 + (x - 3) (x - 1) + (x - 1)^{2}

Expand the expression

x^{2} - 6 x + 9 + x^{2} - 4 x + 3 + (x - 1)^{2}

Expand the expression

x^{2} - 6 x + 9 + x^{2} - 4 x + 3 + x^{2} - 2 x + 1

Add the terms

3 x^{2} - 6 x + 9 - 4 x + 3 - 2 x + 1

Subtract the terms

3 x^{2} - 12 x + 9 + 3 + 1

Add the numbers

3 x^{2} - 12 x + 13

3 x^{2} - 12 x + 13 = 0

Substitute a= 3,b= - 12 and c= 13 into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{12 \pm ( - 12 ) ^{2} - 4 \times 3 \times 13}{2 \times 3}

Simplify the expression

x = \frac{12 \pm ( - 12 ) ^{2} - 4 \times 3 \times 13}{6}

Simplify the expression

More Steps

Evaluate

(- 12)^{2} - 4 \times 3 \times 13

Multiply the terms

(- 12)^{2} - 156

Rewrite the expression

1 2^{2} - 156

Evaluate the power

144 - 156

Subtract the numbers

- 12

x = \frac{12 \pm - 12}{6}

Simplify the radical expression

More Steps

Evaluate

- 12

Evaluate the power

12 \times - 1

Evaluate the power

12 \times i

Evaluate the power

23 \times i

x = \frac{12 \pm 2 3 \times i}{6}

Separate the equation into 2 possible cases

x = \frac{12 + 2 3 \times i}{6} x = \frac{12 - 2 3 \times i}{6}

Simplify the expression

x = 2 + \frac{3}{3} i x = \frac{12 - 2 3 \times i}{6}

Simplify the expression

x = 2 + \frac{3}{3} i x = 2 - \frac{3}{3} i

x \in \emptyset x = 2 + \frac{3}{3} i x = 2 - \frac{3}{3} i

Find the union

x = 2 + \frac{3}{3} i x = 2 - \frac{3}{3} i

Solution

x_{1} = 2 - \frac{3}{3} i, x_{2} = 2 + \frac{3}{3} i

Alternative Form

x_{1} \approx 2 - 0.57735 i, x_{2} \approx 2 + 0.57735 i

Show Solution