Solve (-10z-8) (9z^2-4z-4)

Question

Simplify the expression

- 90 z^{3} - 32 z^{2} + 72 z + 32

Evaluate

(- 10 z - 8) (9 z^{2} - 4 z - 4)

Apply the distributive property

- 10 z \times 9 z^{2} - (- 10 z \times 4 z) - (- 10 z \times 4) - 8 \times 9 z^{2} - (- 8 \times 4 z) - (- 8 \times 4)

Multiply the terms

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Evaluate

- 10 z \times 9 z^{2}

Multiply the numbers

- 90 z \times z^{2}

Multiply the terms

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Evaluate

z \times z^{2}

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

z^{1 + 2}

Add the numbers

z^{3}

- 90 z^{3}

- 90 z^{3} - (- 10 z \times 4 z) - (- 10 z \times 4) - 8 \times 9 z^{2} - (- 8 \times 4 z) - (- 8 \times 4)

Multiply the terms

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Evaluate

- 10 z \times 4 z

Multiply the numbers

- 40 z \times z

Multiply the terms

- 40 z^{2}

- 90 z^{3} - (- 40 z^{2}) - (- 10 z \times 4) - 8 \times 9 z^{2} - (- 8 \times 4 z) - (- 8 \times 4)

Multiply the numbers

- 90 z^{3} - (- 40 z^{2}) - (- 40 z) - 8 \times 9 z^{2} - (- 8 \times 4 z) - (- 8 \times 4)

Multiply the numbers

- 90 z^{3} - (- 40 z^{2}) - (- 40 z) - 72 z^{2} - (- 8 \times 4 z) - (- 8 \times 4)

Multiply the numbers

- 90 z^{3} - (- 40 z^{2}) - (- 40 z) - 72 z^{2} - (- 32 z) - (- 8 \times 4)

Multiply the numbers

- 90 z^{3} - (- 40 z^{2}) - (- 40 z) - 72 z^{2} - (- 32 z) - (- 32)

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

- 90 z^{3} + 40 z^{2} + 40 z - 72 z^{2} + 32 z + 32

Subtract the terms

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Evaluate

40 z^{2} - 72 z^{2}

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

(40 - 72) z^{2}

Subtract the numbers

- 32 z^{2}

- 90 z^{3} - 32 z^{2} + 40 z + 32 z + 32

Solution

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Evaluate

40 z + 32 z

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

(40 + 32) z

Add the numbers

72 z

- 90 z^{3} - 32 z^{2} + 72 z + 32

Show Solution

Factor the expression

- 2 (5 z + 4) (9 z^{2} - 4 z - 4)

Evaluate

(- 10 z - 8) (9 z^{2} - 4 z - 4)

Solution

- 2 (5 z + 4) (9 z^{2} - 4 z - 4)

Show Solution

Find the roots

z_{1} = - \frac{4}{5}, z_{2} = \frac{2 - 2 10}{9}, z_{3} = \frac{2 + 2 10}{9}

Alternative Form

z_{1} = - 0.8, z_{2} \approx - 0.480506, z_{3} \approx 0.924951

Evaluate

(- 10 z - 8) (9 z^{2} - 4 z - 4)

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

(- 10 z - 8) (9 z^{2} - 4 z - 4) = 0

Change the sign

(10 z + 8) (9 z^{2} - 4 z - 4) = 0

Separate the equation into 2 possible cases

10 z + 8 = 0 9 z^{2} - 4 z - 4 = 0

Solve the equation

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Evaluate

10 z + 8 = 0

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

10 z = 0 - 8

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

10 z = - 8

Divide both sides

\frac{10 z}{10} = \frac{- 8}{10}

Divide the numbers

z = \frac{- 8}{10}

Divide the numbers

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Evaluate

\frac{- 8}{10}

Cancel out the common factor 2

\frac{- 4}{5}

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

- \frac{4}{5}

z = - \frac{4}{5}

z = - \frac{4}{5} 9 z^{2} - 4 z - 4 = 0

Solve the equation

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Evaluate

9 z^{2} - 4 z - 4 = 0

Substitute a= 9,b= - 4 and c= - 4 into the quadratic formula z = \frac{- b \pm b ^{2} - 4 a c}{2 a}

z = \frac{4 \pm ( - 4 ) ^{2} - 4 \times 9 ( - 4 )}{2 \times 9}

Simplify the expression

z = \frac{4 \pm ( - 4 ) ^{2} - 4 \times 9 ( - 4 )}{18}

Simplify the expression

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Evaluate

(- 4)^{2} - 4 \times 9 (- 4)

Multiply

(- 4)^{2} - (- 144)

Rewrite the expression

4^{2} - (- 144)

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

4^{2} + 144

Evaluate the power

16 + 144

Add the numbers

160

z = \frac{4 \pm 160}{18}

Simplify the radical expression

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Evaluate

160

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

16 \times 10

Write the number in exponential form with the base of 4

4^{2} \times 10

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

4^{2} \times 10

Reduce the index of the radical and exponent with 2

410

z = \frac{4 \pm 4 10}{18}

Separate the equation into 2 possible cases

z = \frac{4 + 4 10}{18} z = \frac{4 - 4 10}{18}

Simplify the expression

z = \frac{2 + 2 10}{9} z = \frac{4 - 4 10}{18}

Simplify the expression

z = \frac{2 + 2 10}{9} z = \frac{2 - 2 10}{9}

z = - \frac{4}{5} z = \frac{2 + 2 10}{9} z = \frac{2 - 2 10}{9}

Solution

z_{1} = - \frac{4}{5}, z_{2} = \frac{2 - 2 10}{9}, z_{3} = \frac{2 + 2 10}{9}

Alternative Form

z_{1} = - 0.8, z_{2} \approx - 0.480506, z_{3} \approx 0.924951

Show Solution