Solve (7r^2-r-7) ( - 7r-2)

Question

Simplify the expression

- 49 r^{3} - 7 r^{2} + 51 r + 14

Evaluate

(7 r^{2} - r - 7) (- 7 r - 2)

Apply the distributive property

7 r^{2} (- 7 r) - 7 r^{2} \times 2 - r (- 7 r) - (- r \times 2) - 7 (- 7 r) - (- 7 \times 2)

Multiply the terms

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Evaluate

7 r^{2} (- 7 r)

Multiply the numbers

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Evaluate

7 (- 7)

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

- 7 \times 7

Multiply the numbers

- 49

- 49 r^{2} \times r

Multiply the terms

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Evaluate

r^{2} \times r

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

r^{2 + 1}

Add the numbers

r^{3}

- 49 r^{3}

- 49 r^{3} - 7 r^{2} \times 2 - r (- 7 r) - (- r \times 2) - 7 (- 7 r) - (- 7 \times 2)

Multiply the numbers

- 49 r^{3} - 14 r^{2} - r (- 7 r) - (- r \times 2) - 7 (- 7 r) - (- 7 \times 2)

Multiply the terms

More Steps

Evaluate

- r (- 7 r)

Multiply the numbers

7 r \times r

Multiply the terms

7 r^{2}

- 49 r^{3} - 14 r^{2} + 7 r^{2} - (- r \times 2) - 7 (- 7 r) - (- 7 \times 2)

Use the commutative property to reorder the terms

- 49 r^{3} - 14 r^{2} + 7 r^{2} - (- 2 r) - 7 (- 7 r) - (- 7 \times 2)

Multiply the numbers

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Evaluate

- 7 (- 7)

Multiplying or dividing an even number of negative terms equals a positive

7 \times 7

Multiply the numbers

49

- 49 r^{3} - 14 r^{2} + 7 r^{2} - (- 2 r) + 49 r - (- 7 \times 2)

Multiply the numbers

- 49 r^{3} - 14 r^{2} + 7 r^{2} - (- 2 r) + 49 r - (- 14)

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

- 49 r^{3} - 14 r^{2} + 7 r^{2} + 2 r + 49 r + 14

Add the terms

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Evaluate

- 14 r^{2} + 7 r^{2}

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

(- 14 + 7) r^{2}

Add the numbers

- 7 r^{2}

- 49 r^{3} - 7 r^{2} + 2 r + 49 r + 14

Solution

More Steps

Evaluate

2 r + 49 r

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

(2 + 49) r

Add the numbers

51 r

- 49 r^{3} - 7 r^{2} + 51 r + 14

Show Solution

Find the roots

r_{1} = \frac{1 - 197}{14}, r_{2} = - \frac{2}{7}, r_{3} = \frac{1 + 197}{14}

Alternative Form

r_{1} \approx - 0.931119, r_{2} = - 0. \dot{2} 8571 \dot{4}, r_{3} \approx 1.073976

Evaluate

(7 r^{2} - r - 7) (- 7 r - 2)

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

(7 r^{2} - r - 7) (- 7 r - 2) = 0

Separate the equation into 2 possible cases

7 r^{2} - r - 7 = 0 - 7 r - 2 = 0

Solve the equation

More Steps

Evaluate

7 r^{2} - r - 7 = 0

Substitute a= 7,b= - 1 and c= - 7 into the quadratic formula r = \frac{- b \pm b ^{2} - 4 a c}{2 a}

r = \frac{1 \pm ( - 1 ) ^{2} - 4 \times 7 ( - 7 )}{2 \times 7}

Simplify the expression

r = \frac{1 \pm ( - 1 ) ^{2} - 4 \times 7 ( - 7 )}{14}

Simplify the expression

More Steps

Evaluate

(- 1)^{2} - 4 \times 7 (- 7)

Evaluate the power

1 - 4 \times 7 (- 7)

Multiply

1 - (- 196)

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

1 + 196

Add the numbers

197

r = \frac{1 \pm 197}{14}

Separate the equation into 2 possible cases

r = \frac{1 + 197}{14} r = \frac{1 - 197}{14}

r = \frac{1 + 197}{14} r = \frac{1 - 197}{14} - 7 r - 2 = 0

Solve the equation

More Steps

Evaluate

- 7 r - 2 = 0

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

- 7 r = 0 + 2

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

- 7 r = 2

Change the signs on both sides of the equation

7 r = - 2

Divide both sides

\frac{7 r}{7} = \frac{- 2}{7}

Divide the numbers

r = \frac{- 2}{7}

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

r = - \frac{2}{7}

r = \frac{1 + 197}{14} r = \frac{1 - 197}{14} r = - \frac{2}{7}

Solution

r_{1} = \frac{1 - 197}{14}, r_{2} = - \frac{2}{7}, r_{3} = \frac{1 + 197}{14}

Alternative Form

r_{1} \approx - 0.931119, r_{2} = - 0. \dot{2} 8571 \dot{4}, r_{3} \approx 1.073976

Show Solution