Trigonometry Examples

$0.9 z^{2} = 2.4 z - 1.6$

Step 1

Subtract $2.4 z$ from both sides of the equation.

$0.9 z^{2} - 2.4 z = - 1.6$

Step 2

Add $1.6$ to both sides of the equation.

$0.9 z^{2} - 2.4 z + 1.6 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Factor $0.1$ out of $0.9 z^{2} - 2.4 z + 1.6$ .

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Step 3.1.1

Factor $0.1$ out of $0.9 z^{2}$ .

$0.1 (9 z^{2}) - 2.4 z + 1.6 = 0$

Step 3.1.2

Factor $0.1$ out of $- 2.4 z$ .

$0.1 (9 z^{2}) + 0.1 (- 24 z) + 1.6 = 0$

Step 3.1.3

Factor $0.1$ out of $1.6$ .

$0.1 (9 z^{2}) + 0.1 (- 24 z) + 0.1 (16) = 0$

Step 3.1.4

Factor $0.1$ out of $0.1 (9 z^{2}) + 0.1 (- 24 z)$ .

$0.1 (9 z^{2} - 24 z) + 0.1 (16) = 0$

Step 3.1.5

Factor $0.1$ out of $0.1 (9 z^{2} - 24 z) + 0.1 (16)$ .

$0.1 (9 z^{2} - 24 z + 16) = 0$

Step 3.2

Factor using the perfect square rule.

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Step 3.2.1

Rewrite $9 z^{2}$ as ${(3 z)}^{2}$ .

$0.1 ({(3 z)}^{2} - 24 z + 16) = 0$

Step 3.2.2

Rewrite $16$ as $4^{2}$ .

$0.1 ({(3 z)}^{2} - 24 z + 4^{2}) = 0$

Step 3.2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$24 z = 2 \cdot (3 z) \cdot 4$

Step 3.2.4

Rewrite the polynomial.

$0.1 ({(3 z)}^{2} - 2 \cdot (3 z) \cdot 4 + 4^{2}) = 0$

Step 3.2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 3 z$ and $b = 4$ .

$0.1 {(3 z - 4)}^{2} = 0$

Step 4

Divide each term in $0.1 {(3 z - 4)}^{2} = 0$ by $0.1$ and simplify.

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Step 4.1

Divide each term in $0.1 {(3 z - 4)}^{2} = 0$ by $0.1$ .

$\frac{0.1 {(3 z - 4)}^{2}}{0.1} = \frac{0}{0.1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $0.1$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{0.1 {(3 z - 4)}^{2}}{0.1} = \frac{0}{0.1}$

Step 4.2.1.2

Divide ${(3 z - 4)}^{2}$ by $1$ .

${(3 z - 4)}^{2} = \frac{0}{0.1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Divide $0$ by $0.1$ .

${(3 z - 4)}^{2} = 0$

Step 5

Set the $3 z - 4$ equal to $0$ .

$3 z - 4 = 0$

Step 6

Solve for $z$ .

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Step 6.1

Add $4$ to both sides of the equation.

$3 z = 4$

Step 6.2

Divide each term in $3 z = 4$ by $3$ and simplify.

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Step 6.2.1

Divide each term in $3 z = 4$ by $3$ .

$\frac{3 z}{3} = \frac{4}{3}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $3$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{3 z}{3} = \frac{4}{3}$

Step 6.2.2.1.2

Divide $z$ by $1$ .

$z = \frac{4}{3}$