Trigonometry Examples

$x^{2} + 120 = {(3 x + 20)}^{2}$

Step 1

Simplify ${(3 x + 20)}^{2}$ .

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Step 1.1

Rewrite.

$x^{2} + 120 = 0 + 0 + {(3 x + 20)}^{2}$

Step 1.2

Rewrite ${(3 x + 20)}^{2}$ as $(3 x + 20) (3 x + 20)$ .

$x^{2} + 120 = (3 x + 20) (3 x + 20)$

Step 1.3

Expand $(3 x + 20) (3 x + 20)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$x^{2} + 120 = 3 x (3 x + 20) + 20 (3 x + 20)$

Step 1.3.2

Apply the distributive property.

$x^{2} + 120 = 3 x (3 x) + 3 x \cdot 20 + 20 (3 x + 20)$

Step 1.3.3

Apply the distributive property.

$x^{2} + 120 = 3 x (3 x) + 3 x \cdot 20 + 20 (3 x) + 20 \cdot 20$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Rewrite using the commutative property of multiplication.

$x^{2} + 120 = 3 \cdot 3 x \cdot x + 3 x \cdot 20 + 20 (3 x) + 20 \cdot 20$

Step 1.4.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.4.1.2.1

Move $x$ .

$x^{2} + 120 = 3 \cdot 3 (x \cdot x) + 3 x \cdot 20 + 20 (3 x) + 20 \cdot 20$

Step 1.4.1.2.2

Multiply $x$ by $x$ .

$x^{2} + 120 = 3 \cdot 3 x^{2} + 3 x \cdot 20 + 20 (3 x) + 20 \cdot 20$

Step 1.4.1.3

Multiply $3$ by $3$ .

$x^{2} + 120 = 9 x^{2} + 3 x \cdot 20 + 20 (3 x) + 20 \cdot 20$

Step 1.4.1.4

Multiply $20$ by $3$ .

$x^{2} + 120 = 9 x^{2} + 60 x + 20 (3 x) + 20 \cdot 20$

Step 1.4.1.5

Multiply $3$ by $20$ .

$x^{2} + 120 = 9 x^{2} + 60 x + 60 x + 20 \cdot 20$

Step 1.4.1.6

Multiply $20$ by $20$ .

$x^{2} + 120 = 9 x^{2} + 60 x + 60 x + 400$

Step 1.4.2

Add $60 x$ and $60 x$ .

$x^{2} + 120 = 9 x^{2} + 120 x + 400$

Step 2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$9 x^{2} + 120 x + 400 = x^{2} + 120$

Step 3

Move all terms containing $x$ to the left side of the equation.

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Step 3.1

Subtract $x^{2}$ from both sides of the equation.

$9 x^{2} + 120 x + 400 - x^{2} = 120$

Step 3.2

Subtract $x^{2}$ from $9 x^{2}$ .

$8 x^{2} + 120 x + 400 = 120$

Step 4

Subtract $120$ from both sides of the equation.

$8 x^{2} + 120 x + 400 - 120 = 0$

Step 5

Subtract $120$ from $400$ .

$8 x^{2} + 120 x + 280 = 0$

Step 6

Factor $8$ out of $8 x^{2} + 120 x + 280$ .

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Step 6.1

Factor $8$ out of $8 x^{2}$ .

$8 (x^{2}) + 120 x + 280 = 0$

Step 6.2

Factor $8$ out of $120 x$ .

$8 (x^{2}) + 8 (15 x) + 280 = 0$

Step 6.3

Factor $8$ out of $280$ .

$8 x^{2} + 8 (15 x) + 8 \cdot 35 = 0$

Step 6.4

Factor $8$ out of $8 x^{2} + 8 (15 x)$ .

$8 (x^{2} + 15 x) + 8 \cdot 35 = 0$

Step 6.5

Factor $8$ out of $8 (x^{2} + 15 x) + 8 \cdot 35$ .

$8 (x^{2} + 15 x + 35) = 0$

Step 7

Divide each term in $8 (x^{2} + 15 x + 35) = 0$ by $8$ and simplify.

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Step 7.1

Divide each term in $8 (x^{2} + 15 x + 35) = 0$ by $8$ .

$\frac{8 (x^{2} + 15 x + 35)}{8} = \frac{0}{8}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $8$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{8 (x^{2} + 15 x + 35)}{8} = \frac{0}{8}$

Step 7.2.1.2

Divide $x^{2} + 15 x + 35$ by $1$ .

$x^{2} + 15 x + 35 = \frac{0}{8}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $0$ by $8$ .

$x^{2} + 15 x + 35 = 0$

Step 8

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 9

Substitute the values $a = 1$ , $b = 15$ , and $c = 35$ into the quadratic formula and solve for $x$ .

$\frac{- 15 \pm \sqrt{15^{2} - 4 \cdot (1 \cdot 35)}}{2 \cdot 1}$

Step 10

Simplify.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

Raise $15$ to the power of $2$ .

$x = \frac{- 15 \pm \sqrt{225 - 4 \cdot 1 \cdot 35}}{2 \cdot 1}$

Step 10.1.2

Multiply $- 4 \cdot 1 \cdot 35$ .

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Step 10.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 15 \pm \sqrt{225 - 4 \cdot 35}}{2 \cdot 1}$

Step 10.1.2.2

Multiply $- 4$ by $35$ .

$x = \frac{- 15 \pm \sqrt{225 - 140}}{2 \cdot 1}$

Step 10.1.3

Subtract $140$ from $225$ .

$x = \frac{- 15 \pm \sqrt{85}}{2 \cdot 1}$

Step 10.2

Multiply $2$ by $1$ .

$x = \frac{- 15 \pm \sqrt{85}}{2}$

Step 11

The final answer is the combination of both solutions.

$x = - \frac{15 - \sqrt{85}}{2}, - \frac{15 + \sqrt{85}}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{15 - \sqrt{85}}{2}, - \frac{15 + \sqrt{85}}{2}$

Decimal Form:

$x = - 2.89022777 \dots, - 12.10977222 \dots$