Solve (2r+1)(2r^2+2r+3)/3

Question

Simplify the expression

\frac{4 r ^{3} + 6 r ^{2} + 8 r + 3}{3}

Evaluate

(2 r + 1) \times \frac{2 r ^{2} + 2 r + 3}{3}

Multiply the terms

\frac{( 2 r + 1 ) ( 2 r ^{2} + 2 r + 3 )}{3}

Solution

More Steps

Evaluate

(2 r + 1) (2 r^{2} + 2 r + 3)

Apply the distributive property

2 r \times 2 r^{2} + 2 r \times 2 r + 2 r \times 3 + 1 \times 2 r^{2} + 1 \times 2 r + 1 \times 3

Multiply the terms

More Steps

Evaluate

2 r \times 2 r^{2}

Multiply the numbers

4 r \times r^{2}

Multiply the terms

4 r^{3}

4 r^{3} + 2 r \times 2 r + 2 r \times 3 + 1 \times 2 r^{2} + 1 \times 2 r + 1 \times 3

Multiply the terms

More Steps

Evaluate

2 r \times 2 r

Multiply the numbers

4 r \times r

Multiply the terms

4 r^{2}

4 r^{3} + 4 r^{2} + 2 r \times 3 + 1 \times 2 r^{2} + 1 \times 2 r + 1 \times 3

Multiply the numbers

4 r^{3} + 4 r^{2} + 6 r + 1 \times 2 r^{2} + 1 \times 2 r + 1 \times 3

Any expression multiplied by 1 remains the same

4 r^{3} + 4 r^{2} + 6 r + 2 r^{2} + 1 \times 2 r + 1 \times 3

Any expression multiplied by 1 remains the same

4 r^{3} + 4 r^{2} + 6 r + 2 r^{2} + 2 r + 1 \times 3

Any expression multiplied by 1 remains the same

4 r^{3} + 4 r^{2} + 6 r + 2 r^{2} + 2 r + 3

Add the terms

More Steps

Evaluate

4 r^{2} + 2 r^{2}

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

(4 + 2) r^{2}

Add the numbers

6 r^{2}

4 r^{3} + 6 r^{2} + 6 r + 2 r + 3

Add the terms

More Steps

Evaluate

6 r + 2 r

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

(6 + 2) r

Add the numbers

8 r

4 r^{3} + 6 r^{2} + 8 r + 3

\frac{4 r ^{3} + 6 r ^{2} + 8 r + 3}{3}

Show Solution

Find the roots

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

Alternative Form

r_{1} \approx - 0.5 - 1.118034 i, r_{2} \approx - 0.5 + 1.118034 i, r_{3} = - 0.5

Evaluate

(2 r + 1) \times \frac{2 r ^{2} + 2 r + 3}{3}

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

(2 r + 1) \times \frac{2 r ^{2} + 2 r + 3}{3} = 0

Multiply the terms

\frac{( 2 r + 1 ) ( 2 r ^{2} + 2 r + 3 )}{3} = 0

Simplify

(2 r + 1) (2 r^{2} + 2 r + 3) = 0

Separate the equation into 2 possible cases

2 r + 1 = 0 2 r^{2} + 2 r + 3 = 0

Solve the equation

More Steps

Evaluate

2 r + 1 = 0

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

2 r = 0 - 1

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

2 r = - 1

Divide both sides

\frac{2 r}{2} = \frac{- 1}{2}

Divide the numbers

r = \frac{- 1}{2}

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

r = - \frac{1}{2}

r = - \frac{1}{2} 2 r^{2} + 2 r + 3 = 0

Solve the equation

More Steps

Evaluate

2 r^{2} + 2 r + 3 = 0

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

r = \frac{- 2 \pm 2 ^{2} - 4 \times 2 \times 3}{2 \times 2}

Simplify the expression

r = \frac{- 2 \pm 2 ^{2} - 4 \times 2 \times 3}{4}

Simplify the expression

More Steps

Evaluate

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

Multiply the terms

2^{2} - 24

Evaluate the power

4 - 24

Subtract the numbers

- 20

r = \frac{- 2 \pm - 20}{4}

Simplify the radical expression

More Steps

Evaluate

- 20

Evaluate the power

20 \times - 1

Evaluate the power

20 \times i

Evaluate the power

25 \times i

r = \frac{- 2 \pm 2 5 \times i}{4}

Separate the equation into 2 possible cases

r = \frac{- 2 + 2 5 \times i}{4} r = \frac{- 2 - 2 5 \times i}{4}

Simplify the expression

r = - \frac{1}{2} + \frac{5}{2} i r = \frac{- 2 - 2 5 \times i}{4}

Simplify the expression

r = - \frac{1}{2} + \frac{5}{2} i r = - \frac{1}{2} - \frac{5}{2} i

r = - \frac{1}{2} r = - \frac{1}{2} + \frac{5}{2} i r = - \frac{1}{2} - \frac{5}{2} i

Solution

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

Alternative Form

r_{1} \approx - 0.5 - 1.118034 i, r_{2} \approx - 0.5 + 1.118034 i, r_{3} = - 0.5

Show Solution